TROPICAL GEOMETRY
1 "LINEAR EQUATIONS IN TROPICAL GEOMETRY"
Problem 10. Consider the following system of tropical linear equations:
x⊕y= 7
x⊕z= 5
y⊕z= 8
Determine the values of x,y, and zthat satisfy these equations.
Solution 10. a) We start by solving the equation x⊕y= 7. Since ⊕represents the maximum
operation in tropical geometry, the maximum of xand yshould be equal to 7. Therefore, we have:
max(x, y)=7
Given that x, y ∈R, one possible solution is y= 7 and x= 7.
b) Next, we solve the equation x⊕z= 5. This means that the maximum of xand zshould be
equal to 5. Thus, we have:
max(x, z) = 5
Given that x= 7 from part a), we find that z= 5.
c) Lastly, we solve the equation y⊕z= 8, which implies that the maximum of yand zshould
be equal to 8. Hence, we have:
max(y, z)=8
Given that y= 7 from part a), we deduce that z= 8.
Therefore, the values that satisfy the given system of tropical linear equations are x= 7,y= 7,
and z= 8.
2 "COMPUTING TROPICAL VARIETIES WITH NEWTON POLYTOPES"
Problem 4. Consider the tropical variety V(f)defined by the polynomial f(x, y) = max{2x−
y, x +y, 3}.
a) Determine the Newton polytope of f.
b) Find the tropical variety V(f).
Solution 4.
a) The Newton polytope of a polynomial is the convex hull of the exponent vectors of its mono-
mials. In this case, the monomials of fare 2x−y,x+y, and 3.
The exponent vectors of the monomials are {(1,1),(1,1),(0,0)}. The convex hull of these points
is the triangle with vertices at (1,1),(0,0), and (1,0). Therefore, the Newton polytope of fis the
triangle with vertices at these points.
b) To find the tropical variety V(f), we look for points (x, y)at which f(x, y)achieves its minimum
value.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
For f(x, y) = max{2x−y, x +y, 3}to be minimized, each term inside the max function should
individually be minimized. Since we are working in the tropical semiring, minimization is achieved
by taking the minimum of the coefficients of each term.
Thus, f(x, y)reduces to max{2,1,3}, which equals 3. Therefore, the tropical variety V(f)is
the line defined by f(x, y)=3.
In conclusion, the Newton polytope of fis the triangle with vertices at (1,1),(0,0), and (1,0),
and the tropical variety V(f)is the line f(x, y)=3.
3 "COMPUTING TROPICAL INTERSECTION NUMBERS OF ALGEBRAIC CURVES"
Problem 1. Consider the algebraic curve C1:y=x2and C2:y= 2x−1. Compute the tropical
intersection number of C1and C2.
Additional context: In tropical geometry, the tropical intersection number of two algebraic
curves is defined as the number of points at which they intersect in the tropical plane, which is the
max-plus semiring (R∪ {−∞},⊕,⊗).
a) Calculate the tropical intersection point(s) of C1and C2.
b) Determine the tropical intersection number of C1and C2.
Solution 1.
a) To find the tropical intersection point(s) of C1and C2, we need to look for points where the
tropical curves intersect. Since the tropicalization of a polynomial in one variable is just the graph
of the polynomial function with the coefficients replaced by their maximum, we compute the tropical
versions of C1and C2.
The tropicalization of C1:y=x2is the curve given by T rop(C1) : y=max(x⊗x), which
simplifies to T rop(C1) : y=x.
Similarly, the tropicalization of C2:y= 2x−1is T rop(C2) : y=max(2x⊕ −1), which simplifies
to T rop(C2) : y= 2x.
The tropical intersection point is the point where T rop(C1)and T rop(C2)intersect, which in this
case is the point (a, b)where b=a. So, the tropical intersection point is (a, a)or simply the line
y=x.
b) Since the tropical intersection point is a line, the tropical intersection number of C1and C2is
infinity, indicating that the curves C1and C2intersect at infinitely many points in the tropical plane.
4 TROPICAL INTERSECTION THEORY
Problem 2. Consider the tropical polynomials f(x, y) = max{2x+y, x + 3y}and g(x, y) =
max{x+y, 4y−x}.
a) Find the critical points of intersection of fand g.
b) Calculate the value of the tropical intersection f∩gat each critical point.
Solution 2.
a) To find the critical points of intersection of fand g, we need to solve the system of equations
given by setting each pair of linear functions that form fand gequal to each other.
For f(x, y) = max{2x+y, x + 3y}= 2x+y=x+ 3y:2x+y=x+ 3y=⇒x= 2y
For f(x, y) = max{2x+y, x + 3y}=x+ 3y= 4y−x:x+ 3y= 4y−x=⇒x=y
Therefore, the critical points of intersection of fand gare (2t, t)where t∈R.
b) To calculate the value of the tropical intersection f∩gat each critical point (2t, t), we substitute
the values into fand g.
f(2t, t) = max{2(2t) + t, 2t+ 3t}= max{5t, 5t}= 5t
g(2t, t) = max{2t+t, 4t−2t}= max{3t, 2t}= 3t
Hence, the values of the tropical intersection f∩gat the critical points (2t, t)are 5tand 3t.
5 "COMPUTING TROPICAL CONVEX HULLS IN HIGHER DIMENSIONS"
Problem 10. Let’s consider a set of points in 3-dimensional space: P={(2,3,1),(3,1,4),(1,5,2),(4,2,3)}.
Compute the convex hull of these points in the tropical setting.
Solution 10.
a) To compute the tropical convex hull of these points, we need to calculate the tropical sum of
all combinations of three points from the given set P.
b) First, compute the tropical sum of all combinations of three points:
The tropical sum of three points (a, b, c)is given by max(a, b, c)for each coordinate.
For (2,3,1),(3,1,4),(1,5,2):
max(2,3,1) = 3,max(3,1,4) = 4,max(1,5,2) = 5
For (2,3,1),(3,1,4),(4,2,3):
max(2,3,1) = 3,max(3,1,4) = 4,max(4,2,3) = 4
For (2,3,1),(1,5,2),(4,2,3):
max(2,3,1) = 3,max(1,5,2) = 5,max(4,2,3) = 4
For (3,1,4),(1,5,2),(4,2,3):
max(3,1,4) = 4,max(1,5,2) = 5,max(4,2,3) = 4
c) The tropical convex hull consists of the points where at least one of the tropical sums is
achieved. So, the tropical convex hull in this case is {(3,4,5),(3,4,4),(3,5,4),(4,5,4)}.
6 INTERSECTION THEORY IN TROPICAL GEOMETRY
Problem 9. Let P= (0,1), Q = (2,4),and R= (3,2) be points in R2with standard Euclidean
coordinates. Consider the tropical line L=min(x+ 2y, 3x+y).
a) Determine the tropical weight w(P, Q)of the edge between points Pand Q.
b) Find the tropical weight w(Q, R)of the edge between points Qand R.
c) Calculate the tropical weight w(P, R)of the edge between points Pand R.
Solution 9.
a) To find w(P, Q), we substitute the coordinates of Pand Qinto the equation for Land take
the minimum of the resulting expressions:
w(P, Q) = min{0+2·1,3·2+1}= min{2,7}= 2
Therefore, the tropical weight w(P, Q)of the edge between points Pand Qis 2.
b) Similarly, to find w(Q, R), substitute the coordinates of Qand Rinto the equation for Land
take the minimum:
w(Q, R) = min{2+2·4,3·3+4}= min{10,13}= 10
Thus, the tropical weight w(Q, R)of the edge between points Qand Ris 10.
c) Lastly, to calculate w(P, R), substitute the coordinates of Pand Rinto the equation for Land
find the minimum:
w(P, R) = min{0+2·1,3·3+2}= min{2,11}= 2
Hence, the tropical weight w(P, R)of the edge between points Pand Ris 2.
7 "COMPUTING TROPICAL INTERSECTION NUMBERS IN TROPICAL GEOMETRY"
Problem 5. Consider the tropical varieties V1and V2defined by the following polynomial tropical
hypersurfaces in R2:
V1:f1(x, y) = max{2x, y}= 2x∨y
V2:f2(x, y) = max{x+y, 2y}= (x+y)∨2y
Compute the intersection number n(V1∩V2).
Solution 5.
To compute the intersection number n(V1∩V2), we need to find the number of intersection
points of the tropical varieties V1and V2.
The tropical hypersurface V1:f1(x, y) = 2x∨yconsists of two regions separated by the line
2x=y, where the value of f1changes. Similarly, the tropical hypersurface V2:f2(x, y)=(x+y)∨2y
consists of two regions separated by the line x+y= 2y.
To find the intersection points, we need to determine the points where the values of f1and f2
change simultaneously, i.e., where their level sets intersect. This occurs when the lines 2x=y
and x+y= 2ycoincide:
Solving 2x=yand x+y= 2y, we get:
2x=y⇒x=1
2y
x+y= 2y⇒x=y
Setting 1
2y=y, we find y= 0, and hence x= 0 as well. Therefore, the intersection point is
(0,0).
Thus, the intersection number n(V1∩V2)=1since there is only one intersection point.
8 "OPTIMIZATION AND TROPICAL CONVEXITY IN TROPICAL GEOMETRY"
Problem 8. Consider the tropical polynomial f(x, y) = max{2x+y, x + 3y, 4x+y}.
a) Compute f(1,2).
b) Determine the set X={(x, y)∈R2:f(x, y) = 3}.
c) Find the maximum value of f(x, y)over the region defined by x≥0and y≥0.
Solution 8.
a) Plugging in (1,2) into f(x, y) = max{2x+y, x + 3y, 4x+y}gives us f(1,2) = max{2(1) +
2,1 + 3(2),4(1) + 2}= max{4,7,6}= 7.
b) To find the set X={(x, y)∈R2:f(x, y)=3}, we need to determine where the maximum
value of the three terms occurs. Notice that 2x+y= 3 and x+ 3y= 3 when x=y= 1. Also,
4x+y= 3 when x=1
2and y= 2. Thus, X={(1,1),(1
2,2)}.
c) The maximum value of f(x, y)over the region x≥0and y≥0corresponds to the maxima of
each term in the tropical polynomial. Here, each term is a linear function, so to maximize f(x, y), we
set each of these linear functions to be equal to each other. This results in 2x+y=x+3y= 4x+y.
Solving these equations simultaneously gives x=1
2and y= 1, leading to f(1
2,1) = 4 + 1 = 5.
Thus, the maximum value over the region x≥0and y≥0is 5.
9 "POLYHEDRAL SUBDIVISION ALGORITHMS IN TROPICAL GEOMETRY"
Problem 3. Consider the following set of tropical polynomials in variables x, y, z:
f1(x, y, z) = max{2x+y, x + 3y, y +z}
f2(x, y, z) = max{x+ 2y, y + 2z, 3z}
f3(x, y, z) = max{x+y+z, x +y, y +z}
a) Find the piecewise linear function h(x, y, z) = max{f1, f2, f3}.
b) Sketch the polyhedral subdivision of R3induced by h(x, y, z).
Solution 3.
a) To find h(x, y, z) = max{f1, f2, f3}, we need to determine which of the given tropical polyno-
mials gives the maximum value at each point (x, y, z).
For h(x, y, z): - At (x, y, z),h(x, y, z)=2x+yif 2x+y≥x+ 3y, y +z. - h(x, y, z) = x+ 3yif
x+ 3y≥2x+y, y +z. - h(x, y, z) = y+zif y+z≥2x+y, x + 3y.
Thus, the piecewise linear function h(x, y, z)is given by:
h(x, y, z) = max{2x+y, x + 3y, y +z}
b) The polyhedral subdivision of R3induced by h(x, y, z)can be sketched by considering the
regions where each of the three tropical polynomials f1, f2, f3is the maximum. This subdivision
consists of several polyhedral cones, each determined by the maximum polynomial in that region.
The resulting polyhedral subdivision may have cones meeting at certain edges or vertices,
depending on the intersections of the regions where different polynomials are maximized. By iden-
tifying these regions and sketching the corresponding cones based on the above calculations, the
full polyhedral subdivision can be visualized in R3.
10 COMPUTING TROPICAL INTERSECTIONS OF HYPERSURFACES
Problem 6. Given the following two tropical hypersurfaces T rop(f1)and T rop(f2)in R2:
f1(x, y) = x⊕y f2(x, y) = 3 ⊕x⊕2y
a) Compute the tropical intersection T rop(f1)∩T rop(f2).
b) Sketch the tropical hypersurfaces T rop(f1)and T rop(f2)in R2, and illustrate their tropical
intersection graphically.
Solution 6.
a) To compute the tropical intersection T rop(f1)∩T rop(f2), we need to find the minimum of f1
and f2at each lattice point in R2.
At each lattice point (x, y), we have: (T rop(f1)∩T rop(f2))(x, y) = min{f1(x, y), f2(x, y)}
Substitute the expressions for f1and f2:(T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}
Now, we compare the tropical terms: (T rop(f1)∩T rop(f2))(x, y) = min{x⊕y, 3⊕x⊕2y}=
min{x+y, 3 + x+ 2y}
Solving for the minimum, we get: (T rop(f1)∩T rop(f2))(x, y) = x+yif 3 + x+ 2y > x +y
(T rop(f1)∩T rop(f2))(x, y) = 3 + x+ 2yif 3 + x+ 2y≤x+y
Now, we determine the regions where each expression is the minimum.
b) The sketch of T rop(f1)and T rop(f2)in R2can be drawn by representing the tropical hy-
persurfaces as piecewise linear functions with slope changes at the breakpoints. The tropical
intersection graphically illustrates the regions where the minimum function changes.
