TROPICAL GEOMETRY AND ALGEBRAIC COMBINATORICS
1 TROPICAL CONVEXITY AND POLYHEDRAL GEOMETRY
Problem 7. Let f(x, y) = min{3x+ 2y, 2x+ 4y, x + 5y}be a tropical polynomial in two variables.
a) Determine the tropical convex hull of the Newton polytope of f(x, y).
b) Calculate the Newton polygon of f(x, y).
c) Find the tropical hypersurface defined by f(x, y) = α, where αis a constant.
Solution 7.
a) The Newton polytope of f(x, y)is the convex hull of the points (1,0),(0,1), and (0,0). The
tropical convex hull of this polytope is the union of the line segments connecting these points.
Therefore, the tropical convex hull of the Newton polytope of f(x, y)is the triangle with vertices
(1,0),(0,1), and (0,0).
b) The Newton polygon of a tropical polynomial is obtained by taking the convex hull of the
support of the polynomial. In this case, the support of f(x, y)is the set {(1,0),(0,1)}. Therefore,
the Newton polygon of f(x, y)is a line segment connecting (1,0) and (0,1).
c) To find the tropical hypersurface defined by f(x, y) = α, we need to consider the piecewise-
linear function defined by the minimum of the linear functions 3x+ 2y,2x+ 4y, and x+ 5yequal to
α.
Solving each linear function equal to αgives:
3x+ 2y=α,
2x+ 4y=α,
x+ 5y=α.
Solving the above system of equations gives the tropical hypersurface defined by f(x, y) = α
as the piecewise-linear function with three line segments.
2 "COMPUTING TROPICAL CONVEX HULLS OF ALGEBRAIC VARIETIES"
Problem 2. Consider the algebraic curve C={(x, y)∈R2:x2+y2= 1}. Compute the tropical
convex hull of C.
Additional context: The tropical convex hull of an algebraic variety is the union of the tropical
hypersurfaces corresponding to each irreducible component of the variety.
a) Determine the tropical convex hull of the algebraic curve C={(x, y)∈R2:x2+y2= 1}.
Solution 2.
a) The tropical convex hull of an algebraic curve is given by the piecewise linear graph of
max(1,|x|)+max(1,|y|)for each point (x, y)∈C. In this case, the tropical convex hull of Cis
described by the function:
f(x, y) = max(1,|x|) + max(1,|y|)
To visualize this, let’s draw the tropical convex hull on the plane.
This graph consists of four linear segments: 1. For x≥1,f(x, y) = x+ 1 2. For x≤ −1,
f(x, y) = −x+ 1 3. For y≥1,f(x, y) = y+ 1 4. For y≤ −1,f(x, y) = −y+ 1
Therefore, the tropical convex hull of the algebraic curve Cis the set of four line segments that
intersect at the points (1,1),(−1,−1),(−1,1), and (1,−1).
This completes the solution to part (a).
3 "VERTEX DECOMPOSITIONS OF TROPICAL POLYTOPES"
Problem 10. Consider the tropical polytope Pin R3defined as the convex hull of the following
points: A(2,1,3),B(−1,4,2),C(0,2,5), and D(3,0,1). What is the vertex decomposition of P?
Solution 10. To find the vertex decomposition of P, we first need to determine the vertices
of the polytope. We start by considering all the possible tropical affine combinations of the given
points.
a) We need to find the vertices of P. Let’s set up the equation for the tropical affine combination:
Let V=λ1A+λ2B+λ3C+λ4Dbe a point on P.
The vertices are obtained by solving the following system of equations:
2λ1−λ2+λ3+ 3λ4= 0
λ1+ 4λ2+ 2λ3= 0
3λ1+ 2λ2+ 5λ3+λ4= 0
We can solve this system of equations to find the vertex points.
b) Solving the system of equations, we find that the vertices of Pare V1(1,1,2),V2(0,2,2),
V3(1,2,2), and V4(1,2,1).
c) The vertex decomposition of Pis given by the set of vertices {V1, V2, V3, V4}.
4 "MAXIMUM LIKELIHOOD ESTIMATION IN TROPICAL ALGEBRAIC COMBINATORICS"
Problem 1. Consider the following data points: {(2,7),(3,5),(4,3)}. We want to fit a tropical
polynomial f(x) = a⊗x⊕bto this data using the maximum likelihood estimation method. Find the
values of aand bthat maximize the likelihood of the data.
Solution 1.
a) To find the likelihood of the data given the model f(x) = a⊗x⊕b, we need to calculate
the likelihood of each data point under the model. The likelihood of a data point (xi, yi)under the
model is given by f(xi)⊖yi.
For the data point (2,7):
f(2) ⊖7=(a⊗2) ⊕b⊖7
For the data point (3,5):
f(3) ⊖5=(a⊗3) ⊕b⊖5
For the data point (4,3):
f(4) ⊖3=(a⊗4) ⊕b⊖3
b) The maximum likelihood estimation aims to find the values of aand bthat minimize the
difference between the model’s prediction and the observed data. Thus, we want to find aand b
that minimize the sum of squares of the differences:
Likelihood =X
i
[(a⊗xi)⊕b−yi]2
Expanding this expression and plugging in the data points, we get:
Likelihood = ((2a)⊕b−7)2+ ((3a)⊕b−5)2+ ((4a)⊕b−3)2
To minimize this expression, we differentiate with respect to aand b, set the derivatives to 0,
and solve for aand b.
c) Solving the system of equations, we find a= 2 and b= 3. Therefore, the tropical polynomial
that maximizes the likelihood of the data is f(x) = 2 ⊗x⊕3.
5 "POLYHEDRAL REALIZATIONS OF TROPICAL VARIETIES"
Problem 3. Consider the tropical variety Vdefined by the following sets of tropical polynomials:
f1(x, y) = max(3x+y, x + 2y), f2(x, y) = max(x+y, 2x−y)
a) Determine the critical points of the tropical variety V.
b) Find the polyhedral complex that realizes the tropical variety V.
Solution 3.
a) To determine the critical points of the tropical variety V, we need to find where the tropical
polynomials f1and f2have non-unique maximum values.
For f1(x, y) = max(3x+y, x + 2y), we set 3x+y=x+ 2yto find the critical points. Solving this
system of equations gives x=y. The corresponding value of f1at this critical point is f1(x, y) =
max(4x, 2x) = 4x.
For f2(x, y) = max(x+y, 2x−y), we set x+y= 2x−yto find the critical points. Solving this
system of equations gives y=x. The corresponding value of f2at this critical point is f2(x, y) =
max(2x, 2x) = 2x.
Thus, the critical points of the tropical variety Vare points where x=ywith maximum value 4x
under f1, and points where y=xwith maximum value 2xunder f2.
b) To find the polyhedral complex that realizes the tropical variety V, we construct the Newton
polytope of each tropical polynomial.
The Newton polytope of f1(x, y) = max(3x+y, x + 2y)is given by the convex hull of the points
(3,1) and (1,2), which forms a line segment in the xy-plane connecting these two points.
The Newton polytope of f2(x, y) = max(x+y, 2x−y)is given by the convex hull of the points
(1,1) and (2,−1), which forms a line segment in the xy-plane connecting these two points.
Next, we take the union of these two line segments to obtain the polyhedral complex that realizes
the tropical variety V. Therefore, the polyhedral complex corresponding to the tropical variety Vis
a union of two line segments in the xy-plane.
6 "OPTIMAL LINEAR PROGRAMMING IN TROPICAL CONVEX GEOMETRY"
Problem 9. Consider the following system of tropical linear inequalities in variables x,y, and
z:
2x⊕y⊕z≤3
3x⊕2y⊕z≤4
x⊕z≤2
Solve the system and find the optimal values of x,y, and zthat satisfy all the inequalities.
Solution 9. To solve the given system of tropical linear inequalities, we need to minimize the
right-hand side of each inequality and determine the optimal values of x,y, and zthat satisfy all
the inequalities.
a) For the first inequality, minimizing the right-hand side 3gives 2⊕0⊕1=2. Thus, the first
inequality becomes 2x⊕y⊕z≤2.
b) For the second inequality, minimizing the right-hand side 4gives 3⊕0⊕1=3. Thus, the
second inequality becomes 3x⊕2y⊕z≤3.
c) For the third inequality, minimizing the right-hand side 2gives 0⊕0=0. Thus, the third
inequality remains the same as x⊕z≤2.
Now, we have the simplified system of inequalities:
2x⊕y⊕z≤2
3x⊕2y⊕z≤3
x⊕z≤2
To find the optimal values of x,y, and z, we solve these inequalities simultaneously.
From the third inequality x⊕z≤2, we can obtain x≤2and z≤2.
Substitute z≤2into the first inequality 2x⊕y⊕z≤2, we get 2x⊕y≤2. Since x≤2and
2⊕0 = 2, this simplifies to 2=2x⊕y.
Similarly, substitute z≤2into the second inequality 3x⊕2y⊕z≤3, we have 3x⊕2y≤3.
Since x≤2and y≥0,3⊕0=3, we get that 3=3x⊕2y.
Solving 2=2x⊕yand 3=3x⊕2ysimultaneously gives x= 1,y= 0, and z= 2 as the optimal
values that satisfy all the inequalities.
Therefore, the optimal values for x,y, and zare x= 1,y= 0, and z= 2 in the given system.
I. Problem: Consider the tropical polynomial f(x, y) = max{3x+ 2y, x + 4y, 2x+ 2y}in R2.
a) Compute the minimum of f(x, y).
b) Find the points (x, y)where the minimum is attained.
II. Solution:
a) To find the minimum of f(x, y), we look for the minimum value among the three terms inside
the max function. Since the max function takes the maximum value, the minimum of a tropical
polynomial is the minimum of the coefficients.
Therefore, the minimum of f(x, y)is min{3,1,2}= 1.
b) To find the points (x, y)where the minimum is attained, we set each of the terms inside the
max function to the minimum value found in part (a) and solve for xand y.
When 3x+ 2y= 1, we get 3x+ 2y= 1 ⇒3x+ 2y= 1.
When x+ 4y= 1, we have x+ 4y= 1 ⇒x+ 4y= 1.
And when 2x+ 2y= 1,2x+ 2y= 1 ⇒2x+ 2y= 1.
Solving these systems of linear equations, we find that the minimum value of 1 is attained at
the points (x, y) = (1/5,2/5),(1/6,1/6),and (1/3,0).
7 ENUMERATIVE PROBLEMS IN TROPICAL INTERSECTION THEORY
Problem 4. Consider the following tropical hypersurfaces in R2:
H1:x⊕y= 1, H2: 2x⊕2y= 3, H3:x⊕2y= 2.
a) Find all tropical points of intersection of H1,H2, and H3.
b) Calculate the number of bounded regions in R2formed by these tropical hypersurfaces.
c) Determine the tropical intersection multiplicity of the tropical hypersurfaces H1,H2, and H3
at the tropical points of intersection.
Solution 4. a) To find the tropical points of intersection, we need to solve the system of tropical
equations x⊕y= 1,2x⊕2y= 3, and x⊕2y= 2. Solving each pair of equations gives:
x⊕y= 1 ⇒x= 1, y = 0,
2x⊕2y= 3 ⇒x= 2, y =1
2,
x⊕2y= 2 ⇒x= 2, y =1
2.
Hence, the tropical points of intersection are (1,0) and (2,1
2).
b) To determine the number of bounded regions in R2, we need to analyze the convex hull
formed by the tropical points of intersection. The points (1,0) and (2,1
2)lie on the same line x=
2−1
2y. Thus, the tropical hypersurfaces partition R2into 4 bounded regions.
c) Finally, to find the tropical intersection multiplicity, we count the number of times a bounded
region is bounded by the tropical hypersurfaces at the tropical points of intersection. At (1,0),
H1and H3intersect transversely, contributing intersection multiplicity 1 each. At (2,1
2), all three
hypersurfaces intersect, resulting in multiplicity 2.
Therefore, the tropical intersection multiplicity of the tropical hypersurfaces H1,H2and H3at
the tropical points of intersection is 1 + 1 + 2 = 4.
8 "WEIGHTED LAPLACIANS IN TROPICAL GEOMETRY"
Problem 6. Consider the weighted Laplacian of a graph Ggiven by the formula Lweighted(G) =
D−A, where Ais the adjacency matrix of Gand Dis the diagonal matrix containing the weighted
degrees of the vertices.
Let Gbe the graph shown below with edge weights indicated.
1 2
34
3
2 1
2
a) Calculate the adjacency matrix Afor graph G.
b) Calculate the weighted degrees of the vertices and write them in a diagonal matrix D.
c) Find the weighted Laplacian matrix Lweighted(G)for graph G.
Solution 6.
a) The adjacency matrix Afor graph Gis given by the following matrix:
A=
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
b) The weighted degrees of the vertices are the sum of the weights of the edges incident on
each vertex. Therefore, the diagonal matrix Dis:
D=diag(1 + 2 + 3,2+3+1,3+1+2,2 + 2) = diag(6,6,6,4) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
c) The weighted Laplacian matrix Lweighted(G)is then calculated as Lweighted(G) = D−A:
Lweighted(G) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
−
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
=
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
Therefore, the weighted Laplacian matrix for graph Gis
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
.
9 NUMERICAL PROBLEMS
Problem 1. Let v= (3,1,5) and w= (2,4,1) be two vectors in R3. Compute the tropical dot
product ⟨v, w⟩trop.
Problem 2. Consider the matrix M=
231
415
324
. Find the tropical rank of M.
Problem 3. Let A=2 4
5 3and B=3 1
2 4be two matrices. Compute the tropical eigen-
values of the matrix product AB.
10 SOLUTIONS
Solution 1. We compute the tropical dot product ⟨v, w⟩trop by taking the minimum of the entries in
each corresponding position:
⟨v, w⟩trop = (min(3,2),min(1,4),min(5,1)) = (2,1,1).
Therefore, ⟨v, w⟩trop = (2,1,1).
Solution 2. To find the tropical rank of the matrix M, we perform the tropical Gaussian elimi-
nation:
Step 1: Subtract the minimum entry in each row from all entries in that row.
0 1 0
3 0 4
1 0 3
Step 2: Subtract the minimum entry in each column from all entries in that column.
0 1 0
3 0 3
1 0 2
Step 3: Repeat Step 1 and Step 2 until it reaches a tropical zero matrix.
010
302
101
⇒
010
201
100
⇒
010
100
000
The rank of Min tropical algebra is 2.
Solution 3. To compute the tropical eigenvalues of the matrix product AB, we first calculate
AB:
AB =2 4
5 33 1
2 4=16 14
9 13.
The tropical eigenvalues of AB are obtained by taking the maximum entry in each correspond-
ing position of the matrix:
Max Eigenvalue 1 = max(16,14) = 16
Max Eigenvalue 2 = max(9,13) = 13.
Therefore, the tropical eigenvalues of AB are 16 and 13.
Therefore, the tropical convex hull of the algebraic curve Cis the set of four line segments that
intersect at the points (1,1),(−1,−1),(−1,1), and (1,−1).
This completes the solution to part (a).
3 "VERTEX DECOMPOSITIONS OF TROPICAL POLYTOPES"
Problem 10. Consider the tropical polytope Pin R3defined as the convex hull of the following
points: A(2,1,3),B(−1,4,2),C(0,2,5), and D(3,0,1). What is the vertex decomposition of P?
Solution 10. To find the vertex decomposition of P, we first need to determine the vertices
of the polytope. We start by considering all the possible tropical affine combinations of the given
points.
a) We need to find the vertices of P. Let’s set up the equation for the tropical affine combination:
Let V=λ1A+λ2B+λ3C+λ4Dbe a point on P.
The vertices are obtained by solving the following system of equations:
2λ1−λ2+λ3+ 3λ4= 0
λ1+ 4λ2+ 2λ3= 0
3λ1+ 2λ2+ 5λ3+λ4= 0
We can solve this system of equations to find the vertex points.
b) Solving the system of equations, we find that the vertices of Pare V1(1,1,2),V2(0,2,2),
V3(1,2,2), and V4(1,2,1).
c) The vertex decomposition of Pis given by the set of vertices {V1, V2, V3, V4}.
4 "MAXIMUM LIKELIHOOD ESTIMATION IN TROPICAL ALGEBRAIC COMBINATORICS"
Problem 1. Consider the following data points: {(2,7),(3,5),(4,3)}. We want to fit a tropical
polynomial f(x) = a⊗x⊕bto this data using the maximum likelihood estimation method. Find the
values of aand bthat maximize the likelihood of the data.
Solution 1.
a) To find the likelihood of the data given the model f(x) = a⊗x⊕b, we need to calculate
the likelihood of each data point under the model. The likelihood of a data point (xi, yi)under the
model is given by f(xi)⊖yi.
For the data point (2,7):
f(2) ⊖7=(a⊗2) ⊕b⊖7
For the data point (3,5):
f(3) ⊖5=(a⊗3) ⊕b⊖5
For the data point (4,3):
f(4) ⊖3=(a⊗4) ⊕b⊖3
b) The maximum likelihood estimation aims to find the values of aand bthat minimize the
difference between the model’s prediction and the observed data. Thus, we want to find aand b
that minimize the sum of squares of the differences:
Likelihood =X
i
[(a⊗xi)⊕b−yi]2
Expanding this expression and plugging in the data points, we get:
Likelihood = ((2a)⊕b−7)2+ ((3a)⊕b−5)2+ ((4a)⊕b−3)2
To minimize this expression, we differentiate with respect to aand b, set the derivatives to 0,
and solve for aand b.
c) Solving the system of equations, we find a= 2 and b= 3. Therefore, the tropical polynomial
that maximizes the likelihood of the data is f(x) = 2 ⊗x⊕3.
5 "POLYHEDRAL REALIZATIONS OF TROPICAL VARIETIES"
Problem 3. Consider the tropical variety Vdefined by the following sets of tropical polynomials:
f1(x, y) = max(3x+y, x + 2y), f2(x, y) = max(x+y, 2x−y)
a) Determine the critical points of the tropical variety V.
b) Find the polyhedral complex that realizes the tropical variety V.
Solution 3.
a) To determine the critical points of the tropical variety V, we need to find where the tropical
polynomials f1and f2have non-unique maximum values.
For f1(x, y) = max(3x+y, x + 2y), we set 3x+y=x+ 2yto find the critical points. Solving this
system of equations gives x=y. The corresponding value of f1at this critical point is f1(x, y) =
max(4x, 2x) = 4x.
For f2(x, y) = max(x+y, 2x−y), we set x+y= 2x−yto find the critical points. Solving this
system of equations gives y=x. The corresponding value of f2at this critical point is f2(x, y) =
max(2x, 2x) = 2x.
Thus, the critical points of the tropical variety Vare points where x=ywith maximum value 4x
under f1, and points where y=xwith maximum value 2xunder f2.
b) To find the polyhedral complex that realizes the tropical variety V, we construct the Newton
polytope of each tropical polynomial.
The Newton polytope of f1(x, y) = max(3x+y, x + 2y)is given by the convex hull of the points
(3,1) and (1,2), which forms a line segment in the xy-plane connecting these two points.
The Newton polytope of f2(x, y) = max(x+y, 2x−y)is given by the convex hull of the points
(1,1) and (2,−1), which forms a line segment in the xy-plane connecting these two points.
Next, we take the union of these two line segments to obtain the polyhedral complex that realizes
the tropical variety V. Therefore, the polyhedral complex corresponding to the tropical variety Vis
a union of two line segments in the xy-plane.
6 "OPTIMAL LINEAR PROGRAMMING IN TROPICAL CONVEX GEOMETRY"
Problem 9. Consider the following system of tropical linear inequalities in variables x,y, and
z:
2x⊕y⊕z≤3
3x⊕2y⊕z≤4
x⊕z≤2
Solve the system and find the optimal values of x,y, and zthat satisfy all the inequalities.
Solution 9. To solve the given system of tropical linear inequalities, we need to minimize the
right-hand side of each inequality and determine the optimal values of x,y, and zthat satisfy all
the inequalities.
a) For the first inequality, minimizing the right-hand side 3gives 2⊕0⊕1=2. Thus, the first
inequality becomes 2x⊕y⊕z≤2.
b) For the second inequality, minimizing the right-hand side 4gives 3⊕0⊕1=3. Thus, the
second inequality becomes 3x⊕2y⊕z≤3.
c) For the third inequality, minimizing the right-hand side 2gives 0⊕0=0. Thus, the third
inequality remains the same as x⊕z≤2.
Now, we have the simplified system of inequalities:
2x⊕y⊕z≤2
3x⊕2y⊕z≤3
x⊕z≤2
To find the optimal values of x,y, and z, we solve these inequalities simultaneously.
From the third inequality x⊕z≤2, we can obtain x≤2and z≤2.
Substitute z≤2into the first inequality 2x⊕y⊕z≤2, we get 2x⊕y≤2. Since x≤2and
2⊕0 = 2, this simplifies to 2=2x⊕y.
Similarly, substitute z≤2into the second inequality 3x⊕2y⊕z≤3, we have 3x⊕2y≤3.
Since x≤2and y≥0,3⊕0=3, we get that 3=3x⊕2y.
Solving 2=2x⊕yand 3=3x⊕2ysimultaneously gives x= 1,y= 0, and z= 2 as the optimal
values that satisfy all the inequalities.
Therefore, the optimal values for x,y, and zare x= 1,y= 0, and z= 2 in the given system.
I. Problem: Consider the tropical polynomial f(x, y) = max{3x+ 2y, x + 4y, 2x+ 2y}in R2.
a) Compute the minimum of f(x, y).
b) Find the points (x, y)where the minimum is attained.
II. Solution:
a) To find the minimum of f(x, y), we look for the minimum value among the three terms inside
the max function. Since the max function takes the maximum value, the minimum of a tropical
polynomial is the minimum of the coefficients.
Therefore, the minimum of f(x, y)is min{3,1,2}= 1.
b) To find the points (x, y)where the minimum is attained, we set each of the terms inside the
max function to the minimum value found in part (a) and solve for xand y.
When 3x+ 2y= 1, we get 3x+ 2y= 1 ⇒3x+ 2y= 1.
When x+ 4y= 1, we have x+ 4y= 1 ⇒x+ 4y= 1.
And when 2x+ 2y= 1,2x+ 2y= 1 ⇒2x+ 2y= 1.
Solving these systems of linear equations, we find that the minimum value of 1 is attained at
the points (x, y) = (1/5,2/5),(1/6,1/6),and (1/3,0).
7 ENUMERATIVE PROBLEMS IN TROPICAL INTERSECTION THEORY
Problem 4. Consider the following tropical hypersurfaces in R2:
H1:x⊕y= 1, H2: 2x⊕2y= 3, H3:x⊕2y= 2.
a) Find all tropical points of intersection of H1,H2, and H3.
b) Calculate the number of bounded regions in R2formed by these tropical hypersurfaces.
c) Determine the tropical intersection multiplicity of the tropical hypersurfaces H1,H2, and H3
at the tropical points of intersection.
Solution 4. a) To find the tropical points of intersection, we need to solve the system of tropical
equations x⊕y= 1,2x⊕2y= 3, and x⊕2y= 2. Solving each pair of equations gives:
x⊕y= 1 ⇒x= 1, y = 0,
2x⊕2y= 3 ⇒x= 2, y =1
2,
x⊕2y= 2 ⇒x= 2, y =1
2.
Hence, the tropical points of intersection are (1,0) and (2,1
2).
b) To determine the number of bounded regions in R2, we need to analyze the convex hull
formed by the tropical points of intersection. The points (1,0) and (2,1
2)lie on the same line x=
2−1
2y. Thus, the tropical hypersurfaces partition R2into 4 bounded regions.
c) Finally, to find the tropical intersection multiplicity, we count the number of times a bounded
region is bounded by the tropical hypersurfaces at the tropical points of intersection. At (1,0),
H1and H3intersect transversely, contributing intersection multiplicity 1 each. At (2,1
2), all three
hypersurfaces intersect, resulting in multiplicity 2.
Therefore, the tropical intersection multiplicity of the tropical hypersurfaces H1,H2and H3at
the tropical points of intersection is 1 + 1 + 2 = 4.
8 "WEIGHTED LAPLACIANS IN TROPICAL GEOMETRY"
Problem 6. Consider the weighted Laplacian of a graph Ggiven by the formula Lweighted(G) =
D−A, where Ais the adjacency matrix of Gand Dis the diagonal matrix containing the weighted
degrees of the vertices.
Let Gbe the graph shown below with edge weights indicated.
1 2
34
3
2 1
2
a) Calculate the adjacency matrix Afor graph G.
b) Calculate the weighted degrees of the vertices and write them in a diagonal matrix D.
c) Find the weighted Laplacian matrix Lweighted(G)for graph G.
Solution 6.
a) The adjacency matrix Afor graph Gis given by the following matrix:
A=
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
b) The weighted degrees of the vertices are the sum of the weights of the edges incident on
each vertex. Therefore, the diagonal matrix Dis:
D=diag(1 + 2 + 3,2+3+1,3+1+2,2 + 2) = diag(6,6,6,4) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
c) The weighted Laplacian matrix Lweighted(G)is then calculated as Lweighted(G) = D−A:
Lweighted(G) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
−
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
=
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
Therefore, the weighted Laplacian matrix for graph Gis
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
.
9 NUMERICAL PROBLEMS
Problem 1. Let v= (3,1,5) and w= (2,4,1) be two vectors in R3. Compute the tropical dot
product ⟨v, w⟩trop.
Problem 2. Consider the matrix M=
231
415
324
. Find the tropical rank of M.
Problem 3. Let A=2 4
5 3and B=3 1
2 4be two matrices. Compute the tropical eigen-
values of the matrix product AB.
10 SOLUTIONS
Solution 1. We compute the tropical dot product ⟨v, w⟩trop by taking the minimum of the entries in
each corresponding position:
⟨v, w⟩trop = (min(3,2),min(1,4),min(5,1)) = (2,1,1).
Therefore, ⟨v, w⟩trop = (2,1,1).
Solution 2. To find the tropical rank of the matrix M, we perform the tropical Gaussian elimi-
nation:
Step 1: Subtract the minimum entry in each row from all entries in that row.
0 1 0
3 0 4
1 0 3
Step 2: Subtract the minimum entry in each column from all entries in that column.
0 1 0
3 0 3
1 0 2
Step 3: Repeat Step 1 and Step 2 until it reaches a tropical zero matrix.
010
302
101
⇒
010
201
100
⇒
010
100
000
The rank of Min tropical algebra is 2.
Solution 3. To compute the tropical eigenvalues of the matrix product AB, we first calculate
AB:
AB =2 4
5 33 1
2 4=16 14
9 13.
The tropical eigenvalues of AB are obtained by taking the maximum entry in each correspond-
ing position of the matrix:
Max Eigenvalue 1 = max(16,14) = 16
Max Eigenvalue 2 = max(9,13) = 13.
Therefore, the tropical eigenvalues of AB are 16 and 13.
Therefore, the tropical convex hull of the algebraic curve Cis the set of four line segments that
intersect at the points (1,1),(−1,−1),(−1,1), and (1,−1).
This completes the solution to part (a).
3 "VERTEX DECOMPOSITIONS OF TROPICAL POLYTOPES"
Problem 10. Consider the tropical polytope Pin R3defined as the convex hull of the following
points: A(2,1,3),B(−1,4,2),C(0,2,5), and D(3,0,1). What is the vertex decomposition of P?
Solution 10. To find the vertex decomposition of P, we first need to determine the vertices
of the polytope. We start by considering all the possible tropical affine combinations of the given
points.
a) We need to find the vertices of P. Let’s set up the equation for the tropical affine combination:
Let V=λ1A+λ2B+λ3C+λ4Dbe a point on P.
The vertices are obtained by solving the following system of equations:
2λ1−λ2+λ3+ 3λ4= 0
λ1+ 4λ2+ 2λ3= 0
3λ1+ 2λ2+ 5λ3+λ4= 0
We can solve this system of equations to find the vertex points.
b) Solving the system of equations, we find that the vertices of Pare V1(1,1,2),V2(0,2,2),
V3(1,2,2), and V4(1,2,1).
c) The vertex decomposition of Pis given by the set of vertices {V1, V2, V3, V4}.
4 "MAXIMUM LIKELIHOOD ESTIMATION IN TROPICAL ALGEBRAIC COMBINATORICS"
Problem 1. Consider the following data points: {(2,7),(3,5),(4,3)}. We want to fit a tropical
polynomial f(x) = a⊗x⊕bto this data using the maximum likelihood estimation method. Find the
values of aand bthat maximize the likelihood of the data.
Solution 1.
a) To find the likelihood of the data given the model f(x) = a⊗x⊕b, we need to calculate
the likelihood of each data point under the model. The likelihood of a data point (xi, yi)under the
model is given by f(xi)⊖yi.
For the data point (2,7):
f(2) ⊖7=(a⊗2) ⊕b⊖7
For the data point (3,5):
f(3) ⊖5=(a⊗3) ⊕b⊖5
For the data point (4,3):
f(4) ⊖3=(a⊗4) ⊕b⊖3
b) The maximum likelihood estimation aims to find the values of aand bthat minimize the
difference between the model’s prediction and the observed data. Thus, we want to find aand b
that minimize the sum of squares of the differences:
Likelihood =X
i
[(a⊗xi)⊕b−yi]2
Expanding this expression and plugging in the data points, we get:
Likelihood = ((2a)⊕b−7)2+ ((3a)⊕b−5)2+ ((4a)⊕b−3)2
To minimize this expression, we differentiate with respect to aand b, set the derivatives to 0,
and solve for aand b.
c) Solving the system of equations, we find a= 2 and b= 3. Therefore, the tropical polynomial
that maximizes the likelihood of the data is f(x) = 2 ⊗x⊕3.
5 "POLYHEDRAL REALIZATIONS OF TROPICAL VARIETIES"
Problem 3. Consider the tropical variety Vdefined by the following sets of tropical polynomials:
f1(x, y) = max(3x+y, x + 2y), f2(x, y) = max(x+y, 2x−y)
a) Determine the critical points of the tropical variety V.
b) Find the polyhedral complex that realizes the tropical variety V.
Solution 3.
a) To determine the critical points of the tropical variety V, we need to find where the tropical
polynomials f1and f2have non-unique maximum values.
For f1(x, y) = max(3x+y, x + 2y), we set 3x+y=x+ 2yto find the critical points. Solving this
system of equations gives x=y. The corresponding value of f1at this critical point is f1(x, y) =
max(4x, 2x) = 4x.
For f2(x, y) = max(x+y, 2x−y), we set x+y= 2x−yto find the critical points. Solving this
system of equations gives y=x. The corresponding value of f2at this critical point is f2(x, y) =
max(2x, 2x) = 2x.
Thus, the critical points of the tropical variety Vare points where x=ywith maximum value 4x
under f1, and points where y=xwith maximum value 2xunder f2.
b) To find the polyhedral complex that realizes the tropical variety V, we construct the Newton
polytope of each tropical polynomial.
The Newton polytope of f1(x, y) = max(3x+y, x + 2y)is given by the convex hull of the points
(3,1) and (1,2), which forms a line segment in the xy-plane connecting these two points.
The Newton polytope of f2(x, y) = max(x+y, 2x−y)is given by the convex hull of the points
(1,1) and (2,−1), which forms a line segment in the xy-plane connecting these two points.
Next, we take the union of these two line segments to obtain the polyhedral complex that realizes
the tropical variety V. Therefore, the polyhedral complex corresponding to the tropical variety Vis
a union of two line segments in the xy-plane.
6 "OPTIMAL LINEAR PROGRAMMING IN TROPICAL CONVEX GEOMETRY"
Problem 9. Consider the following system of tropical linear inequalities in variables x,y, and
z:
2x⊕y⊕z≤3
3x⊕2y⊕z≤4
x⊕z≤2
Solve the system and find the optimal values of x,y, and zthat satisfy all the inequalities.
Solution 9. To solve the given system of tropical linear inequalities, we need to minimize the
right-hand side of each inequality and determine the optimal values of x,y, and zthat satisfy all
the inequalities.
a) For the first inequality, minimizing the right-hand side 3gives 2⊕0⊕1=2. Thus, the first
inequality becomes 2x⊕y⊕z≤2.
b) For the second inequality, minimizing the right-hand side 4gives 3⊕0⊕1=3. Thus, the
second inequality becomes 3x⊕2y⊕z≤3.
c) For the third inequality, minimizing the right-hand side 2gives 0⊕0=0. Thus, the third
inequality remains the same as x⊕z≤2.
Now, we have the simplified system of inequalities:
2x⊕y⊕z≤2
3x⊕2y⊕z≤3
x⊕z≤2
To find the optimal values of x,y, and z, we solve these inequalities simultaneously.
From the third inequality x⊕z≤2, we can obtain x≤2and z≤2.
Substitute z≤2into the first inequality 2x⊕y⊕z≤2, we get 2x⊕y≤2. Since x≤2and
2⊕0 = 2, this simplifies to 2=2x⊕y.
Similarly, substitute z≤2into the second inequality 3x⊕2y⊕z≤3, we have 3x⊕2y≤3.
Since x≤2and y≥0,3⊕0=3, we get that 3=3x⊕2y.
Solving 2=2x⊕yand 3=3x⊕2ysimultaneously gives x= 1,y= 0, and z= 2 as the optimal
values that satisfy all the inequalities.
Therefore, the optimal values for x,y, and zare x= 1,y= 0, and z= 2 in the given system.
I. Problem: Consider the tropical polynomial f(x, y) = max{3x+ 2y, x + 4y, 2x+ 2y}in R2.
a) Compute the minimum of f(x, y).
b) Find the points (x, y)where the minimum is attained.
II. Solution:
a) To find the minimum of f(x, y), we look for the minimum value among the three terms inside
the max function. Since the max function takes the maximum value, the minimum of a tropical
polynomial is the minimum of the coefficients.
Therefore, the minimum of f(x, y)is min{3,1,2}= 1.
b) To find the points (x, y)where the minimum is attained, we set each of the terms inside the
max function to the minimum value found in part (a) and solve for xand y.
When 3x+ 2y= 1, we get 3x+ 2y= 1 ⇒3x+ 2y= 1.
When x+ 4y= 1, we have x+ 4y= 1 ⇒x+ 4y= 1.
And when 2x+ 2y= 1,2x+ 2y= 1 ⇒2x+ 2y= 1.
Solving these systems of linear equations, we find that the minimum value of 1 is attained at
the points (x, y) = (1/5,2/5),(1/6,1/6),and (1/3,0).
7 ENUMERATIVE PROBLEMS IN TROPICAL INTERSECTION THEORY
Problem 4. Consider the following tropical hypersurfaces in R2:
H1:x⊕y= 1, H2: 2x⊕2y= 3, H3:x⊕2y= 2.
a) Find all tropical points of intersection of H1,H2, and H3.
b) Calculate the number of bounded regions in R2formed by these tropical hypersurfaces.
c) Determine the tropical intersection multiplicity of the tropical hypersurfaces H1,H2, and H3
at the tropical points of intersection.
Solution 4. a) To find the tropical points of intersection, we need to solve the system of tropical
equations x⊕y= 1,2x⊕2y= 3, and x⊕2y= 2. Solving each pair of equations gives:
x⊕y= 1 ⇒x= 1, y = 0,
2x⊕2y= 3 ⇒x= 2, y =1
2,
x⊕2y= 2 ⇒x= 2, y =1
2.
Hence, the tropical points of intersection are (1,0) and (2,1
2).
b) To determine the number of bounded regions in R2, we need to analyze the convex hull
formed by the tropical points of intersection. The points (1,0) and (2,1
2)lie on the same line x=
2−1
2y. Thus, the tropical hypersurfaces partition R2into 4 bounded regions.
c) Finally, to find the tropical intersection multiplicity, we count the number of times a bounded
region is bounded by the tropical hypersurfaces at the tropical points of intersection. At (1,0),
H1and H3intersect transversely, contributing intersection multiplicity 1 each. At (2,1
2), all three
hypersurfaces intersect, resulting in multiplicity 2.
Therefore, the tropical intersection multiplicity of the tropical hypersurfaces H1,H2and H3at
the tropical points of intersection is 1 + 1 + 2 = 4.
8 "WEIGHTED LAPLACIANS IN TROPICAL GEOMETRY"
Problem 6. Consider the weighted Laplacian of a graph Ggiven by the formula Lweighted(G) =
D−A, where Ais the adjacency matrix of Gand Dis the diagonal matrix containing the weighted
degrees of the vertices.
Let Gbe the graph shown below with edge weights indicated.
1 2
34
3
2 1
2
a) Calculate the adjacency matrix Afor graph G.
b) Calculate the weighted degrees of the vertices and write them in a diagonal matrix D.
c) Find the weighted Laplacian matrix Lweighted(G)for graph G.
Solution 6.
a) The adjacency matrix Afor graph Gis given by the following matrix:
A=
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
b) The weighted degrees of the vertices are the sum of the weights of the edges incident on
each vertex. Therefore, the diagonal matrix Dis:
D=diag(1 + 2 + 3,2+3+1,3+1+2,2 + 2) = diag(6,6,6,4) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
c) The weighted Laplacian matrix Lweighted(G)is then calculated as Lweighted(G) = D−A:
Lweighted(G) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
−
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
=
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
Therefore, the weighted Laplacian matrix for graph Gis
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
.
9 NUMERICAL PROBLEMS
Problem 1. Let v= (3,1,5) and w= (2,4,1) be two vectors in R3. Compute the tropical dot
product ⟨v, w⟩trop.
Problem 2. Consider the matrix M=
231
415
324
. Find the tropical rank of M.
Problem 3. Let A=2 4
5 3and B=3 1
2 4be two matrices. Compute the tropical eigen-
values of the matrix product AB.
10 SOLUTIONS
Solution 1. We compute the tropical dot product ⟨v, w⟩trop by taking the minimum of the entries in
each corresponding position:
⟨v, w⟩trop = (min(3,2),min(1,4),min(5,1)) = (2,1,1).
Therefore, ⟨v, w⟩trop = (2,1,1).
Solution 2. To find the tropical rank of the matrix M, we perform the tropical Gaussian elimi-
nation:
Step 1: Subtract the minimum entry in each row from all entries in that row.
0 1 0
3 0 4
1 0 3
Step 2: Subtract the minimum entry in each column from all entries in that column.
0 1 0
3 0 3
1 0 2
Step 3: Repeat Step 1 and Step 2 until it reaches a tropical zero matrix.
010
302
101
⇒
010
201
100
⇒
010
100
000
The rank of Min tropical algebra is 2.
Solution 3. To compute the tropical eigenvalues of the matrix product AB, we first calculate
AB:
AB =2 4
5 33 1
2 4=16 14
9 13.
The tropical eigenvalues of AB are obtained by taking the maximum entry in each correspond-
ing position of the matrix:
Max Eigenvalue 1 = max(16,14) = 16
Max Eigenvalue 2 = max(9,13) = 13.
Therefore, the tropical eigenvalues of AB are 16 and 13.
Therefore, the tropical convex hull of the algebraic curve Cis the set of four line segments that
intersect at the points (1,1),(−1,−1),(−1,1), and (1,−1).
This completes the solution to part (a).
3 "VERTEX DECOMPOSITIONS OF TROPICAL POLYTOPES"
Problem 10. Consider the tropical polytope Pin R3defined as the convex hull of the following
points: A(2,1,3),B(−1,4,2),C(0,2,5), and D(3,0,1). What is the vertex decomposition of P?
Solution 10. To find the vertex decomposition of P, we first need to determine the vertices
of the polytope. We start by considering all the possible tropical affine combinations of the given
points.
a) We need to find the vertices of P. Let’s set up the equation for the tropical affine combination:
Let V=λ1A+λ2B+λ3C+λ4Dbe a point on P.
The vertices are obtained by solving the following system of equations:
2λ1−λ2+λ3+ 3λ4= 0
λ1+ 4λ2+ 2λ3= 0
3λ1+ 2λ2+ 5λ3+λ4= 0
We can solve this system of equations to find the vertex points.
b) Solving the system of equations, we find that the vertices of Pare V1(1,1,2),V2(0,2,2),
V3(1,2,2), and V4(1,2,1).
c) The vertex decomposition of Pis given by the set of vertices {V1, V2, V3, V4}.
4 "MAXIMUM LIKELIHOOD ESTIMATION IN TROPICAL ALGEBRAIC COMBINATORICS"
Problem 1. Consider the following data points: {(2,7),(3,5),(4,3)}. We want to fit a tropical
polynomial f(x) = a⊗x⊕bto this data using the maximum likelihood estimation method. Find the
values of aand bthat maximize the likelihood of the data.
Solution 1.
a) To find the likelihood of the data given the model f(x) = a⊗x⊕b, we need to calculate
the likelihood of each data point under the model. The likelihood of a data point (xi, yi)under the
model is given by f(xi)⊖yi.
For the data point (2,7):
f(2) ⊖7=(a⊗2) ⊕b⊖7
For the data point (3,5):
f(3) ⊖5=(a⊗3) ⊕b⊖5
For the data point (4,3):
f(4) ⊖3=(a⊗4) ⊕b⊖3
b) The maximum likelihood estimation aims to find the values of aand bthat minimize the
difference between the model’s prediction and the observed data. Thus, we want to find aand b
that minimize the sum of squares of the differences:
Likelihood =X
i
[(a⊗xi)⊕b−yi]2
Expanding this expression and plugging in the data points, we get:
Likelihood = ((2a)⊕b−7)2+ ((3a)⊕b−5)2+ ((4a)⊕b−3)2
To minimize this expression, we differentiate with respect to aand b, set the derivatives to 0,
and solve for aand b.
c) Solving the system of equations, we find a= 2 and b= 3. Therefore, the tropical polynomial
that maximizes the likelihood of the data is f(x) = 2 ⊗x⊕3.
5 "POLYHEDRAL REALIZATIONS OF TROPICAL VARIETIES"
Problem 3. Consider the tropical variety Vdefined by the following sets of tropical polynomials:
f1(x, y) = max(3x+y, x + 2y), f2(x, y) = max(x+y, 2x−y)
a) Determine the critical points of the tropical variety V.
b) Find the polyhedral complex that realizes the tropical variety V.
Solution 3.
a) To determine the critical points of the tropical variety V, we need to find where the tropical
polynomials f1and f2have non-unique maximum values.
For f1(x, y) = max(3x+y, x + 2y), we set 3x+y=x+ 2yto find the critical points. Solving this
system of equations gives x=y. The corresponding value of f1at this critical point is f1(x, y) =
max(4x, 2x) = 4x.
For f2(x, y) = max(x+y, 2x−y), we set x+y= 2x−yto find the critical points. Solving this
system of equations gives y=x. The corresponding value of f2at this critical point is f2(x, y) =
max(2x, 2x) = 2x.
Thus, the critical points of the tropical variety Vare points where x=ywith maximum value 4x
under f1, and points where y=xwith maximum value 2xunder f2.
b) To find the polyhedral complex that realizes the tropical variety V, we construct the Newton
polytope of each tropical polynomial.
The Newton polytope of f1(x, y) = max(3x+y, x + 2y)is given by the convex hull of the points
(3,1) and (1,2), which forms a line segment in the xy-plane connecting these two points.
The Newton polytope of f2(x, y) = max(x+y, 2x−y)is given by the convex hull of the points
(1,1) and (2,−1), which forms a line segment in the xy-plane connecting these two points.
Next, we take the union of these two line segments to obtain the polyhedral complex that realizes
the tropical variety V. Therefore, the polyhedral complex corresponding to the tropical variety Vis
a union of two line segments in the xy-plane.
6 "OPTIMAL LINEAR PROGRAMMING IN TROPICAL CONVEX GEOMETRY"
Problem 9. Consider the following system of tropical linear inequalities in variables x,y, and
z:
2x⊕y⊕z≤3
3x⊕2y⊕z≤4
x⊕z≤2
Solve the system and find the optimal values of x,y, and zthat satisfy all the inequalities.
Solution 9. To solve the given system of tropical linear inequalities, we need to minimize the
right-hand side of each inequality and determine the optimal values of x,y, and zthat satisfy all
the inequalities.
a) For the first inequality, minimizing the right-hand side 3gives 2⊕0⊕1=2. Thus, the first
inequality becomes 2x⊕y⊕z≤2.
b) For the second inequality, minimizing the right-hand side 4gives 3⊕0⊕1=3. Thus, the
second inequality becomes 3x⊕2y⊕z≤3.
c) For the third inequality, minimizing the right-hand side 2gives 0⊕0=0. Thus, the third
inequality remains the same as x⊕z≤2.
Now, we have the simplified system of inequalities:
2x⊕y⊕z≤2
3x⊕2y⊕z≤3
x⊕z≤2
To find the optimal values of x,y, and z, we solve these inequalities simultaneously.
From the third inequality x⊕z≤2, we can obtain x≤2and z≤2.
Substitute z≤2into the first inequality 2x⊕y⊕z≤2, we get 2x⊕y≤2. Since x≤2and
2⊕0 = 2, this simplifies to 2=2x⊕y.
Similarly, substitute z≤2into the second inequality 3x⊕2y⊕z≤3, we have 3x⊕2y≤3.
Since x≤2and y≥0,3⊕0=3, we get that 3=3x⊕2y.
Solving 2=2x⊕yand 3=3x⊕2ysimultaneously gives x= 1,y= 0, and z= 2 as the optimal
values that satisfy all the inequalities.
Therefore, the optimal values for x,y, and zare x= 1,y= 0, and z= 2 in the given system.
I. Problem: Consider the tropical polynomial f(x, y) = max{3x+ 2y, x + 4y, 2x+ 2y}in R2.
a) Compute the minimum of f(x, y).
b) Find the points (x, y)where the minimum is attained.
II. Solution:
a) To find the minimum of f(x, y), we look for the minimum value among the three terms inside
the max function. Since the max function takes the maximum value, the minimum of a tropical
polynomial is the minimum of the coefficients.
Therefore, the minimum of f(x, y)is min{3,1,2}= 1.
b) To find the points (x, y)where the minimum is attained, we set each of the terms inside the
max function to the minimum value found in part (a) and solve for xand y.
When 3x+ 2y= 1, we get 3x+ 2y= 1 ⇒3x+ 2y= 1.
When x+ 4y= 1, we have x+ 4y= 1 ⇒x+ 4y= 1.
And when 2x+ 2y= 1,2x+ 2y= 1 ⇒2x+ 2y= 1.
Solving these systems of linear equations, we find that the minimum value of 1 is attained at
the points (x, y) = (1/5,2/5),(1/6,1/6),and (1/3,0).
7 ENUMERATIVE PROBLEMS IN TROPICAL INTERSECTION THEORY
Problem 4. Consider the following tropical hypersurfaces in R2:
H1:x⊕y= 1, H2: 2x⊕2y= 3, H3:x⊕2y= 2.
a) Find all tropical points of intersection of H1,H2, and H3.
b) Calculate the number of bounded regions in R2formed by these tropical hypersurfaces.
c) Determine the tropical intersection multiplicity of the tropical hypersurfaces H1,H2, and H3
at the tropical points of intersection.
Solution 4. a) To find the tropical points of intersection, we need to solve the system of tropical
equations x⊕y= 1,2x⊕2y= 3, and x⊕2y= 2. Solving each pair of equations gives:
x⊕y= 1 ⇒x= 1, y = 0,
2x⊕2y= 3 ⇒x= 2, y =1
2,
x⊕2y= 2 ⇒x= 2, y =1
2.
Hence, the tropical points of intersection are (1,0) and (2,1
2).
b) To determine the number of bounded regions in R2, we need to analyze the convex hull
formed by the tropical points of intersection. The points (1,0) and (2,1
2)lie on the same line x=
2−1
2y. Thus, the tropical hypersurfaces partition R2into 4 bounded regions.
c) Finally, to find the tropical intersection multiplicity, we count the number of times a bounded
region is bounded by the tropical hypersurfaces at the tropical points of intersection. At (1,0),
H1and H3intersect transversely, contributing intersection multiplicity 1 each. At (2,1
2), all three
hypersurfaces intersect, resulting in multiplicity 2.
Therefore, the tropical intersection multiplicity of the tropical hypersurfaces H1,H2and H3at
the tropical points of intersection is 1 + 1 + 2 = 4.
8 "WEIGHTED LAPLACIANS IN TROPICAL GEOMETRY"
Problem 6. Consider the weighted Laplacian of a graph Ggiven by the formula Lweighted(G) =
D−A, where Ais the adjacency matrix of Gand Dis the diagonal matrix containing the weighted
degrees of the vertices.
Let Gbe the graph shown below with edge weights indicated.
1 2
34
3
2 1
2
a) Calculate the adjacency matrix Afor graph G.
b) Calculate the weighted degrees of the vertices and write them in a diagonal matrix D.
c) Find the weighted Laplacian matrix Lweighted(G)for graph G.
Solution 6.
a) The adjacency matrix Afor graph Gis given by the following matrix:
A=
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
b) The weighted degrees of the vertices are the sum of the weights of the edges incident on
each vertex. Therefore, the diagonal matrix Dis:
D=diag(1 + 2 + 3,2+3+1,3+1+2,2 + 2) = diag(6,6,6,4) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
c) The weighted Laplacian matrix Lweighted(G)is then calculated as Lweighted(G) = D−A:
Lweighted(G) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
−
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
=
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
Therefore, the weighted Laplacian matrix for graph Gis
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
.
9 NUMERICAL PROBLEMS
Problem 1. Let v= (3,1,5) and w= (2,4,1) be two vectors in R3. Compute the tropical dot
product ⟨v, w⟩trop.
Problem 2. Consider the matrix M=
231
415
324
. Find the tropical rank of M.
Problem 3. Let A=2 4
5 3and B=3 1
2 4be two matrices. Compute the tropical eigen-
values of the matrix product AB.
10 SOLUTIONS
Solution 1. We compute the tropical dot product ⟨v, w⟩trop by taking the minimum of the entries in
each corresponding position:
⟨v, w⟩trop = (min(3,2),min(1,4),min(5,1)) = (2,1,1).
Therefore, ⟨v, w⟩trop = (2,1,1).
Solution 2. To find the tropical rank of the matrix M, we perform the tropical Gaussian elimi-
nation:
Step 1: Subtract the minimum entry in each row from all entries in that row.
0 1 0
3 0 4
1 0 3
Step 2: Subtract the minimum entry in each column from all entries in that column.
0 1 0
3 0 3
1 0 2
Step 3: Repeat Step 1 and Step 2 until it reaches a tropical zero matrix.
010
302
101
⇒
010
201
100
⇒
010
100
000
The rank of Min tropical algebra is 2.
Solution 3. To compute the tropical eigenvalues of the matrix product AB, we first calculate
AB:
AB =2 4
5 33 1
2 4=16 14
9 13.
The tropical eigenvalues of AB are obtained by taking the maximum entry in each correspond-
ing position of the matrix:
Max Eigenvalue 1 = max(16,14) = 16
Max Eigenvalue 2 = max(9,13) = 13.
Therefore, the tropical eigenvalues of AB are 16 and 13.
Therefore, the tropical convex hull of the algebraic curve Cis the set of four line segments that
intersect at the points (1,1),(−1,−1),(−1,1), and (1,−1).
This completes the solution to part (a).
3 "VERTEX DECOMPOSITIONS OF TROPICAL POLYTOPES"
Problem 10. Consider the tropical polytope Pin R3defined as the convex hull of the following
points: A(2,1,3),B(−1,4,2),C(0,2,5), and D(3,0,1). What is the vertex decomposition of P?
Solution 10. To find the vertex decomposition of P, we first need to determine the vertices
of the polytope. We start by considering all the possible tropical affine combinations of the given
points.
a) We need to find the vertices of P. Let’s set up the equation for the tropical affine combination:
Let V=λ1A+λ2B+λ3C+λ4Dbe a point on P.
The vertices are obtained by solving the following system of equations:
2λ1−λ2+λ3+ 3λ4= 0
λ1+ 4λ2+ 2λ3= 0
3λ1+ 2λ2+ 5λ3+λ4= 0
We can solve this system of equations to find the vertex points.
b) Solving the system of equations, we find that the vertices of Pare V1(1,1,2),V2(0,2,2),
V3(1,2,2), and V4(1,2,1).
c) The vertex decomposition of Pis given by the set of vertices {V1, V2, V3, V4}.
4 "MAXIMUM LIKELIHOOD ESTIMATION IN TROPICAL ALGEBRAIC COMBINATORICS"
Problem 1. Consider the following data points: {(2,7),(3,5),(4,3)}. We want to fit a tropical
polynomial f(x) = a⊗x⊕bto this data using the maximum likelihood estimation method. Find the
values of aand bthat maximize the likelihood of the data.
Solution 1.
a) To find the likelihood of the data given the model f(x) = a⊗x⊕b, we need to calculate
the likelihood of each data point under the model. The likelihood of a data point (xi, yi)under the
model is given by f(xi)⊖yi.
For the data point (2,7):
f(2) ⊖7=(a⊗2) ⊕b⊖7
For the data point (3,5):
f(3) ⊖5=(a⊗3) ⊕b⊖5
For the data point (4,3):
f(4) ⊖3=(a⊗4) ⊕b⊖3
b) The maximum likelihood estimation aims to find the values of aand bthat minimize the
difference between the model’s prediction and the observed data. Thus, we want to find aand b
that minimize the sum of squares of the differences:
Likelihood =X
i
[(a⊗xi)⊕b−yi]2
Expanding this expression and plugging in the data points, we get:
Likelihood = ((2a)⊕b−7)2+ ((3a)⊕b−5)2+ ((4a)⊕b−3)2
To minimize this expression, we differentiate with respect to aand b, set the derivatives to 0,
and solve for aand b.
c) Solving the system of equations, we find a= 2 and b= 3. Therefore, the tropical polynomial
that maximizes the likelihood of the data is f(x) = 2 ⊗x⊕3.
5 "POLYHEDRAL REALIZATIONS OF TROPICAL VARIETIES"
Problem 3. Consider the tropical variety Vdefined by the following sets of tropical polynomials:
f1(x, y) = max(3x+y, x + 2y), f2(x, y) = max(x+y, 2x−y)
a) Determine the critical points of the tropical variety V.
b) Find the polyhedral complex that realizes the tropical variety V.
Solution 3.
a) To determine the critical points of the tropical variety V, we need to find where the tropical
polynomials f1and f2have non-unique maximum values.
For f1(x, y) = max(3x+y, x + 2y), we set 3x+y=x+ 2yto find the critical points. Solving this
system of equations gives x=y. The corresponding value of f1at this critical point is f1(x, y) =
max(4x, 2x) = 4x.
For f2(x, y) = max(x+y, 2x−y), we set x+y= 2x−yto find the critical points. Solving this
system of equations gives y=x. The corresponding value of f2at this critical point is f2(x, y) =
max(2x, 2x) = 2x.
Thus, the critical points of the tropical variety Vare points where x=ywith maximum value 4x
under f1, and points where y=xwith maximum value 2xunder f2.
b) To find the polyhedral complex that realizes the tropical variety V, we construct the Newton
polytope of each tropical polynomial.
The Newton polytope of f1(x, y) = max(3x+y, x + 2y)is given by the convex hull of the points
(3,1) and (1,2), which forms a line segment in the xy-plane connecting these two points.
The Newton polytope of f2(x, y) = max(x+y, 2x−y)is given by the convex hull of the points
(1,1) and (2,−1), which forms a line segment in the xy-plane connecting these two points.
Next, we take the union of these two line segments to obtain the polyhedral complex that realizes
the tropical variety V. Therefore, the polyhedral complex corresponding to the tropical variety Vis
a union of two line segments in the xy-plane.
6 "OPTIMAL LINEAR PROGRAMMING IN TROPICAL CONVEX GEOMETRY"
Problem 9. Consider the following system of tropical linear inequalities in variables x,y, and
z:
2x⊕y⊕z≤3
3x⊕2y⊕z≤4
x⊕z≤2
Solve the system and find the optimal values of x,y, and zthat satisfy all the inequalities.
Solution 9. To solve the given system of tropical linear inequalities, we need to minimize the
right-hand side of each inequality and determine the optimal values of x,y, and zthat satisfy all
the inequalities.
a) For the first inequality, minimizing the right-hand side 3gives 2⊕0⊕1=2. Thus, the first
inequality becomes 2x⊕y⊕z≤2.
b) For the second inequality, minimizing the right-hand side 4gives 3⊕0⊕1=3. Thus, the
second inequality becomes 3x⊕2y⊕z≤3.
c) For the third inequality, minimizing the right-hand side 2gives 0⊕0=0. Thus, the third
inequality remains the same as x⊕z≤2.
Now, we have the simplified system of inequalities:
2x⊕y⊕z≤2
3x⊕2y⊕z≤3
x⊕z≤2
To find the optimal values of x,y, and z, we solve these inequalities simultaneously.
From the third inequality x⊕z≤2, we can obtain x≤2and z≤2.
Substitute z≤2into the first inequality 2x⊕y⊕z≤2, we get 2x⊕y≤2. Since x≤2and
2⊕0 = 2, this simplifies to 2=2x⊕y.
Similarly, substitute z≤2into the second inequality 3x⊕2y⊕z≤3, we have 3x⊕2y≤3.
Since x≤2and y≥0,3⊕0=3, we get that 3=3x⊕2y.
Solving 2=2x⊕yand 3=3x⊕2ysimultaneously gives x= 1,y= 0, and z= 2 as the optimal
values that satisfy all the inequalities.
Therefore, the optimal values for x,y, and zare x= 1,y= 0, and z= 2 in the given system.
I. Problem: Consider the tropical polynomial f(x, y) = max{3x+ 2y, x + 4y, 2x+ 2y}in R2.
a) Compute the minimum of f(x, y).
b) Find the points (x, y)where the minimum is attained.
II. Solution:
a) To find the minimum of f(x, y), we look for the minimum value among the three terms inside
the max function. Since the max function takes the maximum value, the minimum of a tropical
polynomial is the minimum of the coefficients.
Therefore, the minimum of f(x, y)is min{3,1,2}= 1.
b) To find the points (x, y)where the minimum is attained, we set each of the terms inside the
max function to the minimum value found in part (a) and solve for xand y.
When 3x+ 2y= 1, we get 3x+ 2y= 1 ⇒3x+ 2y= 1.
When x+ 4y= 1, we have x+ 4y= 1 ⇒x+ 4y= 1.
And when 2x+ 2y= 1,2x+ 2y= 1 ⇒2x+ 2y= 1.
Solving these systems of linear equations, we find that the minimum value of 1 is attained at
the points (x, y) = (1/5,2/5),(1/6,1/6),and (1/3,0).
7 ENUMERATIVE PROBLEMS IN TROPICAL INTERSECTION THEORY
Problem 4. Consider the following tropical hypersurfaces in R2:
H1:x⊕y= 1, H2: 2x⊕2y= 3, H3:x⊕2y= 2.
a) Find all tropical points of intersection of H1,H2, and H3.
b) Calculate the number of bounded regions in R2formed by these tropical hypersurfaces.
c) Determine the tropical intersection multiplicity of the tropical hypersurfaces H1,H2, and H3
at the tropical points of intersection.
Solution 4. a) To find the tropical points of intersection, we need to solve the system of tropical
equations x⊕y= 1,2x⊕2y= 3, and x⊕2y= 2. Solving each pair of equations gives:
x⊕y= 1 ⇒x= 1, y = 0,
2x⊕2y= 3 ⇒x= 2, y =1
2,
x⊕2y= 2 ⇒x= 2, y =1
2.
Hence, the tropical points of intersection are (1,0) and (2,1
2).
b) To determine the number of bounded regions in R2, we need to analyze the convex hull
formed by the tropical points of intersection. The points (1,0) and (2,1
2)lie on the same line x=
2−1
2y. Thus, the tropical hypersurfaces partition R2into 4 bounded regions.
c) Finally, to find the tropical intersection multiplicity, we count the number of times a bounded
region is bounded by the tropical hypersurfaces at the tropical points of intersection. At (1,0),
H1and H3intersect transversely, contributing intersection multiplicity 1 each. At (2,1
2), all three
hypersurfaces intersect, resulting in multiplicity 2.
Therefore, the tropical intersection multiplicity of the tropical hypersurfaces H1,H2and H3at
the tropical points of intersection is 1 + 1 + 2 = 4.
8 "WEIGHTED LAPLACIANS IN TROPICAL GEOMETRY"
Problem 6. Consider the weighted Laplacian of a graph Ggiven by the formula Lweighted(G) =
D−A, where Ais the adjacency matrix of Gand Dis the diagonal matrix containing the weighted
degrees of the vertices.
Let Gbe the graph shown below with edge weights indicated.
1 2
34
3
2 1
2
a) Calculate the adjacency matrix Afor graph G.
b) Calculate the weighted degrees of the vertices and write them in a diagonal matrix D.
c) Find the weighted Laplacian matrix Lweighted(G)for graph G.
Solution 6.
a) The adjacency matrix Afor graph Gis given by the following matrix:
A=
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
b) The weighted degrees of the vertices are the sum of the weights of the edges incident on
each vertex. Therefore, the diagonal matrix Dis:
D=diag(1 + 2 + 3,2+3+1,3+1+2,2 + 2) = diag(6,6,6,4) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
c) The weighted Laplacian matrix Lweighted(G)is then calculated as Lweighted(G) = D−A:
Lweighted(G) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
−
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
=
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
Therefore, the weighted Laplacian matrix for graph Gis
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
.
9 NUMERICAL PROBLEMS
Problem 1. Let v= (3,1,5) and w= (2,4,1) be two vectors in R3. Compute the tropical dot
product ⟨v, w⟩trop.
Problem 2. Consider the matrix M=
231
415
324
. Find the tropical rank of M.
Problem 3. Let A=2 4
5 3and B=3 1
2 4be two matrices. Compute the tropical eigen-
values of the matrix product AB.
10 SOLUTIONS
Solution 1. We compute the tropical dot product ⟨v, w⟩trop by taking the minimum of the entries in
each corresponding position:
⟨v, w⟩trop = (min(3,2),min(1,4),min(5,1)) = (2,1,1).
Therefore, ⟨v, w⟩trop = (2,1,1).
Solution 2. To find the tropical rank of the matrix M, we perform the tropical Gaussian elimi-
nation:
Step 1: Subtract the minimum entry in each row from all entries in that row.
0 1 0
3 0 4
1 0 3
Step 2: Subtract the minimum entry in each column from all entries in that column.
0 1 0
3 0 3
1 0 2
Step 3: Repeat Step 1 and Step 2 until it reaches a tropical zero matrix.
010
302
101
⇒
010
201
100
⇒
010
100
000
The rank of Min tropical algebra is 2.
Solution 3. To compute the tropical eigenvalues of the matrix product AB, we first calculate
AB:
AB =2 4
5 33 1
2 4=16 14
9 13.
The tropical eigenvalues of AB are obtained by taking the maximum entry in each correspond-
ing position of the matrix:
Max Eigenvalue 1 = max(16,14) = 16
Max Eigenvalue 2 = max(9,13) = 13.
Therefore, the tropical eigenvalues of AB are 16 and 13.
Therefore, the tropical convex hull of the algebraic curve Cis the set of four line segments that
intersect at the points (1,1),(−1,−1),(−1,1), and (1,−1).
This completes the solution to part (a).
3 "VERTEX DECOMPOSITIONS OF TROPICAL POLYTOPES"
Problem 10. Consider the tropical polytope Pin R3defined as the convex hull of the following
points: A(2,1,3),B(−1,4,2),C(0,2,5), and D(3,0,1). What is the vertex decomposition of P?
Solution 10. To find the vertex decomposition of P, we first need to determine the vertices
of the polytope. We start by considering all the possible tropical affine combinations of the given
points.
a) We need to find the vertices of P. Let’s set up the equation for the tropical affine combination:
Let V=λ1A+λ2B+λ3C+λ4Dbe a point on P.
The vertices are obtained by solving the following system of equations:
2λ1−λ2+λ3+ 3λ4= 0
λ1+ 4λ2+ 2λ3= 0
3λ1+ 2λ2+ 5λ3+λ4= 0
We can solve this system of equations to find the vertex points.
b) Solving the system of equations, we find that the vertices of Pare V1(1,1,2),V2(0,2,2),
V3(1,2,2), and V4(1,2,1).
c) The vertex decomposition of Pis given by the set of vertices {V1, V2, V3, V4}.
4 "MAXIMUM LIKELIHOOD ESTIMATION IN TROPICAL ALGEBRAIC COMBINATORICS"
Problem 1. Consider the following data points: {(2,7),(3,5),(4,3)}. We want to fit a tropical
polynomial f(x) = a⊗x⊕bto this data using the maximum likelihood estimation method. Find the
values of aand bthat maximize the likelihood of the data.
Solution 1.
a) To find the likelihood of the data given the model f(x) = a⊗x⊕b, we need to calculate
the likelihood of each data point under the model. The likelihood of a data point (xi, yi)under the
model is given by f(xi)⊖yi.
For the data point (2,7):
f(2) ⊖7=(a⊗2) ⊕b⊖7
For the data point (3,5):
f(3) ⊖5=(a⊗3) ⊕b⊖5
For the data point (4,3):
f(4) ⊖3=(a⊗4) ⊕b⊖3
b) The maximum likelihood estimation aims to find the values of aand bthat minimize the
difference between the model’s prediction and the observed data. Thus, we want to find aand b
that minimize the sum of squares of the differences:
Likelihood =X
i
[(a⊗xi)⊕b−yi]2
Expanding this expression and plugging in the data points, we get:
Likelihood = ((2a)⊕b−7)2+ ((3a)⊕b−5)2+ ((4a)⊕b−3)2
To minimize this expression, we differentiate with respect to aand b, set the derivatives to 0,
and solve for aand b.
c) Solving the system of equations, we find a= 2 and b= 3. Therefore, the tropical polynomial
that maximizes the likelihood of the data is f(x) = 2 ⊗x⊕3.
5 "POLYHEDRAL REALIZATIONS OF TROPICAL VARIETIES"
Problem 3. Consider the tropical variety Vdefined by the following sets of tropical polynomials:
f1(x, y) = max(3x+y, x + 2y), f2(x, y) = max(x+y, 2x−y)
a) Determine the critical points of the tropical variety V.
b) Find the polyhedral complex that realizes the tropical variety V.
Solution 3.
a) To determine the critical points of the tropical variety V, we need to find where the tropical
polynomials f1and f2have non-unique maximum values.
For f1(x, y) = max(3x+y, x + 2y), we set 3x+y=x+ 2yto find the critical points. Solving this
system of equations gives x=y. The corresponding value of f1at this critical point is f1(x, y) =
max(4x, 2x) = 4x.
For f2(x, y) = max(x+y, 2x−y), we set x+y= 2x−yto find the critical points. Solving this
system of equations gives y=x. The corresponding value of f2at this critical point is f2(x, y) =
max(2x, 2x) = 2x.
Thus, the critical points of the tropical variety Vare points where x=ywith maximum value 4x
under f1, and points where y=xwith maximum value 2xunder f2.
b) To find the polyhedral complex that realizes the tropical variety V, we construct the Newton
polytope of each tropical polynomial.
The Newton polytope of f1(x, y) = max(3x+y, x + 2y)is given by the convex hull of the points
(3,1) and (1,2), which forms a line segment in the xy-plane connecting these two points.
The Newton polytope of f2(x, y) = max(x+y, 2x−y)is given by the convex hull of the points
(1,1) and (2,−1), which forms a line segment in the xy-plane connecting these two points.
Next, we take the union of these two line segments to obtain the polyhedral complex that realizes
the tropical variety V. Therefore, the polyhedral complex corresponding to the tropical variety Vis
a union of two line segments in the xy-plane.
6 "OPTIMAL LINEAR PROGRAMMING IN TROPICAL CONVEX GEOMETRY"
Problem 9. Consider the following system of tropical linear inequalities in variables x,y, and
z:
2x⊕y⊕z≤3
3x⊕2y⊕z≤4
x⊕z≤2
Solve the system and find the optimal values of x,y, and zthat satisfy all the inequalities.
Solution 9. To solve the given system of tropical linear inequalities, we need to minimize the
right-hand side of each inequality and determine the optimal values of x,y, and zthat satisfy all
the inequalities.
a) For the first inequality, minimizing the right-hand side 3gives 2⊕0⊕1=2. Thus, the first
inequality becomes 2x⊕y⊕z≤2.
b) For the second inequality, minimizing the right-hand side 4gives 3⊕0⊕1=3. Thus, the
second inequality becomes 3x⊕2y⊕z≤3.
c) For the third inequality, minimizing the right-hand side 2gives 0⊕0=0. Thus, the third
inequality remains the same as x⊕z≤2.
Now, we have the simplified system of inequalities:
2x⊕y⊕z≤2
3x⊕2y⊕z≤3
x⊕z≤2
To find the optimal values of x,y, and z, we solve these inequalities simultaneously.
From the third inequality x⊕z≤2, we can obtain x≤2and z≤2.
Substitute z≤2into the first inequality 2x⊕y⊕z≤2, we get 2x⊕y≤2. Since x≤2and
2⊕0 = 2, this simplifies to 2=2x⊕y.
Similarly, substitute z≤2into the second inequality 3x⊕2y⊕z≤3, we have 3x⊕2y≤3.
Since x≤2and y≥0,3⊕0=3, we get that 3=3x⊕2y.
Solving 2=2x⊕yand 3=3x⊕2ysimultaneously gives x= 1,y= 0, and z= 2 as the optimal
values that satisfy all the inequalities.
Therefore, the optimal values for x,y, and zare x= 1,y= 0, and z= 2 in the given system.
I. Problem: Consider the tropical polynomial f(x, y) = max{3x+ 2y, x + 4y, 2x+ 2y}in R2.
a) Compute the minimum of f(x, y).
b) Find the points (x, y)where the minimum is attained.
II. Solution:
a) To find the minimum of f(x, y), we look for the minimum value among the three terms inside
the max function. Since the max function takes the maximum value, the minimum of a tropical
polynomial is the minimum of the coefficients.
Therefore, the minimum of f(x, y)is min{3,1,2}= 1.
b) To find the points (x, y)where the minimum is attained, we set each of the terms inside the
max function to the minimum value found in part (a) and solve for xand y.
When 3x+ 2y= 1, we get 3x+ 2y= 1 ⇒3x+ 2y= 1.
When x+ 4y= 1, we have x+ 4y= 1 ⇒x+ 4y= 1.
And when 2x+ 2y= 1,2x+ 2y= 1 ⇒2x+ 2y= 1.
Solving these systems of linear equations, we find that the minimum value of 1 is attained at
the points (x, y) = (1/5,2/5),(1/6,1/6),and (1/3,0).
7 ENUMERATIVE PROBLEMS IN TROPICAL INTERSECTION THEORY
Problem 4. Consider the following tropical hypersurfaces in R2:
H1:x⊕y= 1, H2: 2x⊕2y= 3, H3:x⊕2y= 2.
a) Find all tropical points of intersection of H1,H2, and H3.
b) Calculate the number of bounded regions in R2formed by these tropical hypersurfaces.
c) Determine the tropical intersection multiplicity of the tropical hypersurfaces H1,H2, and H3
at the tropical points of intersection.
Solution 4. a) To find the tropical points of intersection, we need to solve the system of tropical
equations x⊕y= 1,2x⊕2y= 3, and x⊕2y= 2. Solving each pair of equations gives:
x⊕y= 1 ⇒x= 1, y = 0,
2x⊕2y= 3 ⇒x= 2, y =1
2,
x⊕2y= 2 ⇒x= 2, y =1
2.
Hence, the tropical points of intersection are (1,0) and (2,1
2).
b) To determine the number of bounded regions in R2, we need to analyze the convex hull
formed by the tropical points of intersection. The points (1,0) and (2,1
2)lie on the same line x=
2−1
2y. Thus, the tropical hypersurfaces partition R2into 4 bounded regions.
c) Finally, to find the tropical intersection multiplicity, we count the number of times a bounded
region is bounded by the tropical hypersurfaces at the tropical points of intersection. At (1,0),
H1and H3intersect transversely, contributing intersection multiplicity 1 each. At (2,1
2), all three
hypersurfaces intersect, resulting in multiplicity 2.
Therefore, the tropical intersection multiplicity of the tropical hypersurfaces H1,H2and H3at
the tropical points of intersection is 1 + 1 + 2 = 4.
8 "WEIGHTED LAPLACIANS IN TROPICAL GEOMETRY"
Problem 6. Consider the weighted Laplacian of a graph Ggiven by the formula Lweighted(G) =
D−A, where Ais the adjacency matrix of Gand Dis the diagonal matrix containing the weighted
degrees of the vertices.
Let Gbe the graph shown below with edge weights indicated.
1 2
34
3
2 1
2
a) Calculate the adjacency matrix Afor graph G.
b) Calculate the weighted degrees of the vertices and write them in a diagonal matrix D.
c) Find the weighted Laplacian matrix Lweighted(G)for graph G.
Solution 6.
a) The adjacency matrix Afor graph Gis given by the following matrix:
A=
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
b) The weighted degrees of the vertices are the sum of the weights of the edges incident on
each vertex. Therefore, the diagonal matrix Dis:
D=diag(1 + 2 + 3,2+3+1,3+1+2,2 + 2) = diag(6,6,6,4) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
c) The weighted Laplacian matrix Lweighted(G)is then calculated as Lweighted(G) = D−A:
Lweighted(G) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
−
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
=
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
Therefore, the weighted Laplacian matrix for graph Gis
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
.
9 NUMERICAL PROBLEMS
Problem 1. Let v= (3,1,5) and w= (2,4,1) be two vectors in R3. Compute the tropical dot
product ⟨v, w⟩trop.
Problem 2. Consider the matrix M=
231
415
324
. Find the tropical rank of M.
Problem 3. Let A=2 4
5 3and B=3 1
2 4be two matrices. Compute the tropical eigen-
values of the matrix product AB.
10 SOLUTIONS
Solution 1. We compute the tropical dot product ⟨v, w⟩trop by taking the minimum of the entries in
each corresponding position:
⟨v, w⟩trop = (min(3,2),min(1,4),min(5,1)) = (2,1,1).
Therefore, ⟨v, w⟩trop = (2,1,1).
Solution 2. To find the tropical rank of the matrix M, we perform the tropical Gaussian elimi-
nation:
Step 1: Subtract the minimum entry in each row from all entries in that row.
0 1 0
3 0 4
1 0 3
Step 2: Subtract the minimum entry in each column from all entries in that column.
0 1 0
3 0 3
1 0 2
Step 3: Repeat Step 1 and Step 2 until it reaches a tropical zero matrix.
010
302
101
⇒
010
201
100
⇒
010
100
000
The rank of Min tropical algebra is 2.
Solution 3. To compute the tropical eigenvalues of the matrix product AB, we first calculate
AB:
AB =2 4
5 33 1
2 4=16 14
9 13.
The tropical eigenvalues of AB are obtained by taking the maximum entry in each correspond-
ing position of the matrix:
Max Eigenvalue 1 = max(16,14) = 16
Max Eigenvalue 2 = max(9,13) = 13.
Therefore, the tropical eigenvalues of AB are 16 and 13.
Therefore, the tropical convex hull of the algebraic curve Cis the set of four line segments that
intersect at the points (1,1),(−1,−1),(−1,1), and (1,−1).
This completes the solution to part (a).
3 "VERTEX DECOMPOSITIONS OF TROPICAL POLYTOPES"
Problem 10. Consider the tropical polytope Pin R3defined as the convex hull of the following
points: A(2,1,3),B(−1,4,2),C(0,2,5), and D(3,0,1). What is the vertex decomposition of P?
Solution 10. To find the vertex decomposition of P, we first need to determine the vertices
of the polytope. We start by considering all the possible tropical affine combinations of the given
points.
a) We need to find the vertices of P. Let’s set up the equation for the tropical affine combination:
Let V=λ1A+λ2B+λ3C+λ4Dbe a point on P.
The vertices are obtained by solving the following system of equations:
2λ1−λ2+λ3+ 3λ4= 0
λ1+ 4λ2+ 2λ3= 0
3λ1+ 2λ2+ 5λ3+λ4= 0
We can solve this system of equations to find the vertex points.
b) Solving the system of equations, we find that the vertices of Pare V1(1,1,2),V2(0,2,2),
V3(1,2,2), and V4(1,2,1).
c) The vertex decomposition of Pis given by the set of vertices {V1, V2, V3, V4}.
4 "MAXIMUM LIKELIHOOD ESTIMATION IN TROPICAL ALGEBRAIC COMBINATORICS"
Problem 1. Consider the following data points: {(2,7),(3,5),(4,3)}. We want to fit a tropical
polynomial f(x) = a⊗x⊕bto this data using the maximum likelihood estimation method. Find the
values of aand bthat maximize the likelihood of the data.
Solution 1.
a) To find the likelihood of the data given the model f(x) = a⊗x⊕b, we need to calculate
the likelihood of each data point under the model. The likelihood of a data point (xi, yi)under the
model is given by f(xi)⊖yi.
For the data point (2,7):
f(2) ⊖7=(a⊗2) ⊕b⊖7
For the data point (3,5):
f(3) ⊖5=(a⊗3) ⊕b⊖5
For the data point (4,3):
f(4) ⊖3=(a⊗4) ⊕b⊖3
b) The maximum likelihood estimation aims to find the values of aand bthat minimize the
difference between the model’s prediction and the observed data. Thus, we want to find aand b
that minimize the sum of squares of the differences:
Likelihood =X
i
[(a⊗xi)⊕b−yi]2
Expanding this expression and plugging in the data points, we get:
Likelihood = ((2a)⊕b−7)2+ ((3a)⊕b−5)2+ ((4a)⊕b−3)2
To minimize this expression, we differentiate with respect to aand b, set the derivatives to 0,
and solve for aand b.
c) Solving the system of equations, we find a= 2 and b= 3. Therefore, the tropical polynomial
that maximizes the likelihood of the data is f(x) = 2 ⊗x⊕3.
5 "POLYHEDRAL REALIZATIONS OF TROPICAL VARIETIES"
Problem 3. Consider the tropical variety Vdefined by the following sets of tropical polynomials:
f1(x, y) = max(3x+y, x + 2y), f2(x, y) = max(x+y, 2x−y)
a) Determine the critical points of the tropical variety V.
b) Find the polyhedral complex that realizes the tropical variety V.
Solution 3.
a) To determine the critical points of the tropical variety V, we need to find where the tropical
polynomials f1and f2have non-unique maximum values.
For f1(x, y) = max(3x+y, x + 2y), we set 3x+y=x+ 2yto find the critical points. Solving this
system of equations gives x=y. The corresponding value of f1at this critical point is f1(x, y) =
max(4x, 2x) = 4x.
For f2(x, y) = max(x+y, 2x−y), we set x+y= 2x−yto find the critical points. Solving this
system of equations gives y=x. The corresponding value of f2at this critical point is f2(x, y) =
max(2x, 2x) = 2x.
Thus, the critical points of the tropical variety Vare points where x=ywith maximum value 4x
under f1, and points where y=xwith maximum value 2xunder f2.
b) To find the polyhedral complex that realizes the tropical variety V, we construct the Newton
polytope of each tropical polynomial.
The Newton polytope of f1(x, y) = max(3x+y, x + 2y)is given by the convex hull of the points
(3,1) and (1,2), which forms a line segment in the xy-plane connecting these two points.
The Newton polytope of f2(x, y) = max(x+y, 2x−y)is given by the convex hull of the points
(1,1) and (2,−1), which forms a line segment in the xy-plane connecting these two points.
Next, we take the union of these two line segments to obtain the polyhedral complex that realizes
the tropical variety V. Therefore, the polyhedral complex corresponding to the tropical variety Vis
a union of two line segments in the xy-plane.
6 "OPTIMAL LINEAR PROGRAMMING IN TROPICAL CONVEX GEOMETRY"
Problem 9. Consider the following system of tropical linear inequalities in variables x,y, and
z:
2x⊕y⊕z≤3
3x⊕2y⊕z≤4
x⊕z≤2
Solve the system and find the optimal values of x,y, and zthat satisfy all the inequalities.
Solution 9. To solve the given system of tropical linear inequalities, we need to minimize the
right-hand side of each inequality and determine the optimal values of x,y, and zthat satisfy all
the inequalities.
a) For the first inequality, minimizing the right-hand side 3gives 2⊕0⊕1=2. Thus, the first
inequality becomes 2x⊕y⊕z≤2.
b) For the second inequality, minimizing the right-hand side 4gives 3⊕0⊕1=3. Thus, the
second inequality becomes 3x⊕2y⊕z≤3.
c) For the third inequality, minimizing the right-hand side 2gives 0⊕0=0. Thus, the third
inequality remains the same as x⊕z≤2.
Now, we have the simplified system of inequalities:
2x⊕y⊕z≤2
3x⊕2y⊕z≤3
x⊕z≤2
To find the optimal values of x,y, and z, we solve these inequalities simultaneously.
From the third inequality x⊕z≤2, we can obtain x≤2and z≤2.
Substitute z≤2into the first inequality 2x⊕y⊕z≤2, we get 2x⊕y≤2. Since x≤2and
2⊕0 = 2, this simplifies to 2=2x⊕y.
Similarly, substitute z≤2into the second inequality 3x⊕2y⊕z≤3, we have 3x⊕2y≤3.
Since x≤2and y≥0,3⊕0=3, we get that 3=3x⊕2y.
Solving 2=2x⊕yand 3=3x⊕2ysimultaneously gives x= 1,y= 0, and z= 2 as the optimal
values that satisfy all the inequalities.
Therefore, the optimal values for x,y, and zare x= 1,y= 0, and z= 2 in the given system.
I. Problem: Consider the tropical polynomial f(x, y) = max{3x+ 2y, x + 4y, 2x+ 2y}in R2.
a) Compute the minimum of f(x, y).
b) Find the points (x, y)where the minimum is attained.
II. Solution:
a) To find the minimum of f(x, y), we look for the minimum value among the three terms inside
the max function. Since the max function takes the maximum value, the minimum of a tropical
polynomial is the minimum of the coefficients.
Therefore, the minimum of f(x, y)is min{3,1,2}= 1.
b) To find the points (x, y)where the minimum is attained, we set each of the terms inside the
max function to the minimum value found in part (a) and solve for xand y.
When 3x+ 2y= 1, we get 3x+ 2y= 1 ⇒3x+ 2y= 1.
When x+ 4y= 1, we have x+ 4y= 1 ⇒x+ 4y= 1.
And when 2x+ 2y= 1,2x+ 2y= 1 ⇒2x+ 2y= 1.
Solving these systems of linear equations, we find that the minimum value of 1 is attained at
the points (x, y) = (1/5,2/5),(1/6,1/6),and (1/3,0).
7 ENUMERATIVE PROBLEMS IN TROPICAL INTERSECTION THEORY
Problem 4. Consider the following tropical hypersurfaces in R2:
H1:x⊕y= 1, H2: 2x⊕2y= 3, H3:x⊕2y= 2.
a) Find all tropical points of intersection of H1,H2, and H3.
b) Calculate the number of bounded regions in R2formed by these tropical hypersurfaces.
c) Determine the tropical intersection multiplicity of the tropical hypersurfaces H1,H2, and H3
at the tropical points of intersection.
Solution 4. a) To find the tropical points of intersection, we need to solve the system of tropical
equations x⊕y= 1,2x⊕2y= 3, and x⊕2y= 2. Solving each pair of equations gives:
x⊕y= 1 ⇒x= 1, y = 0,
2x⊕2y= 3 ⇒x= 2, y =1
2,
x⊕2y= 2 ⇒x= 2, y =1
2.
Hence, the tropical points of intersection are (1,0) and (2,1
2).
b) To determine the number of bounded regions in R2, we need to analyze the convex hull
formed by the tropical points of intersection. The points (1,0) and (2,1
2)lie on the same line x=
2−1
2y. Thus, the tropical hypersurfaces partition R2into 4 bounded regions.
c) Finally, to find the tropical intersection multiplicity, we count the number of times a bounded
region is bounded by the tropical hypersurfaces at the tropical points of intersection. At (1,0),
H1and H3intersect transversely, contributing intersection multiplicity 1 each. At (2,1
2), all three
hypersurfaces intersect, resulting in multiplicity 2.
Therefore, the tropical intersection multiplicity of the tropical hypersurfaces H1,H2and H3at
the tropical points of intersection is 1 + 1 + 2 = 4.
8 "WEIGHTED LAPLACIANS IN TROPICAL GEOMETRY"
Problem 6. Consider the weighted Laplacian of a graph Ggiven by the formula Lweighted(G) =
D−A, where Ais the adjacency matrix of Gand Dis the diagonal matrix containing the weighted
degrees of the vertices.
Let Gbe the graph shown below with edge weights indicated.
1 2
34
3
2 1
2
a) Calculate the adjacency matrix Afor graph G.
b) Calculate the weighted degrees of the vertices and write them in a diagonal matrix D.
c) Find the weighted Laplacian matrix Lweighted(G)for graph G.
Solution 6.
a) The adjacency matrix Afor graph Gis given by the following matrix:
A=
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
b) The weighted degrees of the vertices are the sum of the weights of the edges incident on
each vertex. Therefore, the diagonal matrix Dis:
D=diag(1 + 2 + 3,2+3+1,3+1+2,2 + 2) = diag(6,6,6,4) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
c) The weighted Laplacian matrix Lweighted(G)is then calculated as Lweighted(G) = D−A:
Lweighted(G) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
−
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
=
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
Therefore, the weighted Laplacian matrix for graph Gis
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
.
9 NUMERICAL PROBLEMS
Problem 1. Let v= (3,1,5) and w= (2,4,1) be two vectors in R3. Compute the tropical dot
product ⟨v, w⟩trop.
Problem 2. Consider the matrix M=
231
415
324
. Find the tropical rank of M.
Problem 3. Let A=2 4
5 3and B=3 1
2 4be two matrices. Compute the tropical eigen-
values of the matrix product AB.
10 SOLUTIONS
Solution 1. We compute the tropical dot product ⟨v, w⟩trop by taking the minimum of the entries in
each corresponding position:
⟨v, w⟩trop = (min(3,2),min(1,4),min(5,1)) = (2,1,1).
Therefore, ⟨v, w⟩trop = (2,1,1).
Solution 2. To find the tropical rank of the matrix M, we perform the tropical Gaussian elimi-
nation:
Step 1: Subtract the minimum entry in each row from all entries in that row.
0 1 0
3 0 4
1 0 3
Step 2: Subtract the minimum entry in each column from all entries in that column.
0 1 0
3 0 3
1 0 2
Step 3: Repeat Step 1 and Step 2 until it reaches a tropical zero matrix.
010
302
101
⇒
010
201
100
⇒
010
100
000
The rank of Min tropical algebra is 2.
Solution 3. To compute the tropical eigenvalues of the matrix product AB, we first calculate
AB:
AB =2 4
5 33 1
2 4=16 14
9 13.
The tropical eigenvalues of AB are obtained by taking the maximum entry in each correspond-
ing position of the matrix:
Max Eigenvalue 1 = max(16,14) = 16
Max Eigenvalue 2 = max(9,13) = 13.
Therefore, the tropical eigenvalues of AB are 16 and 13.
Therefore, the tropical convex hull of the algebraic curve Cis the set of four line segments that
intersect at the points (1,1),(−1,−1),(−1,1), and (1,−1).
This completes the solution to part (a).
3 "VERTEX DECOMPOSITIONS OF TROPICAL POLYTOPES"
Problem 10. Consider the tropical polytope Pin R3defined as the convex hull of the following
points: A(2,1,3),B(−1,4,2),C(0,2,5), and D(3,0,1). What is the vertex decomposition of P?
Solution 10. To find the vertex decomposition of P, we first need to determine the vertices
of the polytope. We start by considering all the possible tropical affine combinations of the given
points.
a) We need to find the vertices of P. Let’s set up the equation for the tropical affine combination:
Let V=λ1A+λ2B+λ3C+λ4Dbe a point on P.
The vertices are obtained by solving the following system of equations:
2λ1−λ2+λ3+ 3λ4= 0
λ1+ 4λ2+ 2λ3= 0
3λ1+ 2λ2+ 5λ3+λ4= 0
We can solve this system of equations to find the vertex points.
b) Solving the system of equations, we find that the vertices of Pare V1(1,1,2),V2(0,2,2),
V3(1,2,2), and V4(1,2,1).
c) The vertex decomposition of Pis given by the set of vertices {V1, V2, V3, V4}.
4 "MAXIMUM LIKELIHOOD ESTIMATION IN TROPICAL ALGEBRAIC COMBINATORICS"
Problem 1. Consider the following data points: {(2,7),(3,5),(4,3)}. We want to fit a tropical
polynomial f(x) = a⊗x⊕bto this data using the maximum likelihood estimation method. Find the
values of aand bthat maximize the likelihood of the data.
Solution 1.
a) To find the likelihood of the data given the model f(x) = a⊗x⊕b, we need to calculate
the likelihood of each data point under the model. The likelihood of a data point (xi, yi)under the
model is given by f(xi)⊖yi.
For the data point (2,7):
f(2) ⊖7=(a⊗2) ⊕b⊖7
For the data point (3,5):
f(3) ⊖5=(a⊗3) ⊕b⊖5
For the data point (4,3):
f(4) ⊖3=(a⊗4) ⊕b⊖3
b) The maximum likelihood estimation aims to find the values of aand bthat minimize the
difference between the model’s prediction and the observed data. Thus, we want to find aand b
that minimize the sum of squares of the differences:
Likelihood =X
i
[(a⊗xi)⊕b−yi]2
Expanding this expression and plugging in the data points, we get:
Likelihood = ((2a)⊕b−7)2+ ((3a)⊕b−5)2+ ((4a)⊕b−3)2
To minimize this expression, we differentiate with respect to aand b, set the derivatives to 0,
and solve for aand b.
c) Solving the system of equations, we find a= 2 and b= 3. Therefore, the tropical polynomial
that maximizes the likelihood of the data is f(x) = 2 ⊗x⊕3.
5 "POLYHEDRAL REALIZATIONS OF TROPICAL VARIETIES"
Problem 3. Consider the tropical variety Vdefined by the following sets of tropical polynomials:
f1(x, y) = max(3x+y, x + 2y), f2(x, y) = max(x+y, 2x−y)
a) Determine the critical points of the tropical variety V.
b) Find the polyhedral complex that realizes the tropical variety V.
Solution 3.
a) To determine the critical points of the tropical variety V, we need to find where the tropical
polynomials f1and f2have non-unique maximum values.
For f1(x, y) = max(3x+y, x + 2y), we set 3x+y=x+ 2yto find the critical points. Solving this
system of equations gives x=y. The corresponding value of f1at this critical point is f1(x, y) =
max(4x, 2x) = 4x.
For f2(x, y) = max(x+y, 2x−y), we set x+y= 2x−yto find the critical points. Solving this
system of equations gives y=x. The corresponding value of f2at this critical point is f2(x, y) =
max(2x, 2x) = 2x.
Thus, the critical points of the tropical variety Vare points where x=ywith maximum value 4x
under f1, and points where y=xwith maximum value 2xunder f2.
b) To find the polyhedral complex that realizes the tropical variety V, we construct the Newton
polytope of each tropical polynomial.
The Newton polytope of f1(x, y) = max(3x+y, x + 2y)is given by the convex hull of the points
(3,1) and (1,2), which forms a line segment in the xy-plane connecting these two points.
The Newton polytope of f2(x, y) = max(x+y, 2x−y)is given by the convex hull of the points
(1,1) and (2,−1), which forms a line segment in the xy-plane connecting these two points.
Next, we take the union of these two line segments to obtain the polyhedral complex that realizes
the tropical variety V. Therefore, the polyhedral complex corresponding to the tropical variety Vis
a union of two line segments in the xy-plane.
6 "OPTIMAL LINEAR PROGRAMMING IN TROPICAL CONVEX GEOMETRY"
Problem 9. Consider the following system of tropical linear inequalities in variables x,y, and
z:
2x⊕y⊕z≤3
3x⊕2y⊕z≤4
x⊕z≤2
Solve the system and find the optimal values of x,y, and zthat satisfy all the inequalities.
Solution 9. To solve the given system of tropical linear inequalities, we need to minimize the
right-hand side of each inequality and determine the optimal values of x,y, and zthat satisfy all
the inequalities.
a) For the first inequality, minimizing the right-hand side 3gives 2⊕0⊕1=2. Thus, the first
inequality becomes 2x⊕y⊕z≤2.
b) For the second inequality, minimizing the right-hand side 4gives 3⊕0⊕1=3. Thus, the
second inequality becomes 3x⊕2y⊕z≤3.
c) For the third inequality, minimizing the right-hand side 2gives 0⊕0=0. Thus, the third
inequality remains the same as x⊕z≤2.
Now, we have the simplified system of inequalities:
2x⊕y⊕z≤2
3x⊕2y⊕z≤3
x⊕z≤2
To find the optimal values of x,y, and z, we solve these inequalities simultaneously.
From the third inequality x⊕z≤2, we can obtain x≤2and z≤2.
Substitute z≤2into the first inequality 2x⊕y⊕z≤2, we get 2x⊕y≤2. Since x≤2and
2⊕0 = 2, this simplifies to 2=2x⊕y.
Similarly, substitute z≤2into the second inequality 3x⊕2y⊕z≤3, we have 3x⊕2y≤3.
Since x≤2and y≥0,3⊕0=3, we get that 3=3x⊕2y.
Solving 2=2x⊕yand 3=3x⊕2ysimultaneously gives x= 1,y= 0, and z= 2 as the optimal
values that satisfy all the inequalities.
Therefore, the optimal values for x,y, and zare x= 1,y= 0, and z= 2 in the given system.
I. Problem: Consider the tropical polynomial f(x, y) = max{3x+ 2y, x + 4y, 2x+ 2y}in R2.
a) Compute the minimum of f(x, y).
b) Find the points (x, y)where the minimum is attained.
II. Solution:
a) To find the minimum of f(x, y), we look for the minimum value among the three terms inside
the max function. Since the max function takes the maximum value, the minimum of a tropical
polynomial is the minimum of the coefficients.
Therefore, the minimum of f(x, y)is min{3,1,2}= 1.
b) To find the points (x, y)where the minimum is attained, we set each of the terms inside the
max function to the minimum value found in part (a) and solve for xand y.
When 3x+ 2y= 1, we get 3x+ 2y= 1 ⇒3x+ 2y= 1.
When x+ 4y= 1, we have x+ 4y= 1 ⇒x+ 4y= 1.
And when 2x+ 2y= 1,2x+ 2y= 1 ⇒2x+ 2y= 1.
Solving these systems of linear equations, we find that the minimum value of 1 is attained at
the points (x, y) = (1/5,2/5),(1/6,1/6),and (1/3,0).
7 ENUMERATIVE PROBLEMS IN TROPICAL INTERSECTION THEORY
Problem 4. Consider the following tropical hypersurfaces in R2:
H1:x⊕y= 1, H2: 2x⊕2y= 3, H3:x⊕2y= 2.
a) Find all tropical points of intersection of H1,H2, and H3.
b) Calculate the number of bounded regions in R2formed by these tropical hypersurfaces.
c) Determine the tropical intersection multiplicity of the tropical hypersurfaces H1,H2, and H3
at the tropical points of intersection.
Solution 4. a) To find the tropical points of intersection, we need to solve the system of tropical
equations x⊕y= 1,2x⊕2y= 3, and x⊕2y= 2. Solving each pair of equations gives:
x⊕y= 1 ⇒x= 1, y = 0,
2x⊕2y= 3 ⇒x= 2, y =1
2,
x⊕2y= 2 ⇒x= 2, y =1
2.
Hence, the tropical points of intersection are (1,0) and (2,1
2).
b) To determine the number of bounded regions in R2, we need to analyze the convex hull
formed by the tropical points of intersection. The points (1,0) and (2,1
2)lie on the same line x=
2−1
2y. Thus, the tropical hypersurfaces partition R2into 4 bounded regions.
c) Finally, to find the tropical intersection multiplicity, we count the number of times a bounded
region is bounded by the tropical hypersurfaces at the tropical points of intersection. At (1,0),
H1and H3intersect transversely, contributing intersection multiplicity 1 each. At (2,1
2), all three
hypersurfaces intersect, resulting in multiplicity 2.
Therefore, the tropical intersection multiplicity of the tropical hypersurfaces H1,H2and H3at
the tropical points of intersection is 1 + 1 + 2 = 4.
8 "WEIGHTED LAPLACIANS IN TROPICAL GEOMETRY"
Problem 6. Consider the weighted Laplacian of a graph Ggiven by the formula Lweighted(G) =
D−A, where Ais the adjacency matrix of Gand Dis the diagonal matrix containing the weighted
degrees of the vertices.
Let Gbe the graph shown below with edge weights indicated.
1 2
34
3
2 1
2
a) Calculate the adjacency matrix Afor graph G.
b) Calculate the weighted degrees of the vertices and write them in a diagonal matrix D.
c) Find the weighted Laplacian matrix Lweighted(G)for graph G.
Solution 6.
a) The adjacency matrix Afor graph Gis given by the following matrix:
A=
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
b) The weighted degrees of the vertices are the sum of the weights of the edges incident on
each vertex. Therefore, the diagonal matrix Dis:
D=diag(1 + 2 + 3,2+3+1,3+1+2,2 + 2) = diag(6,6,6,4) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
c) The weighted Laplacian matrix Lweighted(G)is then calculated as Lweighted(G) = D−A:
Lweighted(G) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
−
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
=
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
Therefore, the weighted Laplacian matrix for graph Gis
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
.
9 NUMERICAL PROBLEMS
Problem 1. Let v= (3,1,5) and w= (2,4,1) be two vectors in R3. Compute the tropical dot
product ⟨v, w⟩trop.
Problem 2. Consider the matrix M=
231
415
324
. Find the tropical rank of M.
Problem 3. Let A=2 4
5 3and B=3 1
2 4be two matrices. Compute the tropical eigen-
values of the matrix product AB.
10 SOLUTIONS
Solution 1. We compute the tropical dot product ⟨v, w⟩trop by taking the minimum of the entries in
each corresponding position:
⟨v, w⟩trop = (min(3,2),min(1,4),min(5,1)) = (2,1,1).
Therefore, ⟨v, w⟩trop = (2,1,1).
Solution 2. To find the tropical rank of the matrix M, we perform the tropical Gaussian elimi-
nation:
Step 1: Subtract the minimum entry in each row from all entries in that row.
0 1 0
3 0 4
1 0 3
Step 2: Subtract the minimum entry in each column from all entries in that column.
0 1 0
3 0 3
1 0 2
Step 3: Repeat Step 1 and Step 2 until it reaches a tropical zero matrix.
010
302
101
⇒
010
201
100
⇒
010
100
000
The rank of Min tropical algebra is 2.
Solution 3. To compute the tropical eigenvalues of the matrix product AB, we first calculate
AB:
AB =2 4
5 33 1
2 4=16 14
9 13.
The tropical eigenvalues of AB are obtained by taking the maximum entry in each correspond-
ing position of the matrix:
Max Eigenvalue 1 = max(16,14) = 16
Max Eigenvalue 2 = max(9,13) = 13.
Therefore, the tropical eigenvalues of AB are 16 and 13.
Therefore, the tropical convex hull of the algebraic curve Cis the set of four line segments that
intersect at the points (1,1),(−1,−1),(−1,1), and (1,−1).
This completes the solution to part (a).
3 "VERTEX DECOMPOSITIONS OF TROPICAL POLYTOPES"
Problem 10. Consider the tropical polytope Pin R3defined as the convex hull of the following
points: A(2,1,3),B(−1,4,2),C(0,2,5), and D(3,0,1). What is the vertex decomposition of P?
Solution 10. To find the vertex decomposition of P, we first need to determine the vertices
of the polytope. We start by considering all the possible tropical affine combinations of the given
points.
a) We need to find the vertices of P. Let’s set up the equation for the tropical affine combination:
Let V=λ1A+λ2B+λ3C+λ4Dbe a point on P.
The vertices are obtained by solving the following system of equations:
2λ1−λ2+λ3+ 3λ4= 0
λ1+ 4λ2+ 2λ3= 0
3λ1+ 2λ2+ 5λ3+λ4= 0
We can solve this system of equations to find the vertex points.
b) Solving the system of equations, we find that the vertices of Pare V1(1,1,2),V2(0,2,2),
V3(1,2,2), and V4(1,2,1).
c) The vertex decomposition of Pis given by the set of vertices {V1, V2, V3, V4}.
4 "MAXIMUM LIKELIHOOD ESTIMATION IN TROPICAL ALGEBRAIC COMBINATORICS"
Problem 1. Consider the following data points: {(2,7),(3,5),(4,3)}. We want to fit a tropical
polynomial f(x) = a⊗x⊕bto this data using the maximum likelihood estimation method. Find the
values of aand bthat maximize the likelihood of the data.
Solution 1.
a) To find the likelihood of the data given the model f(x) = a⊗x⊕b, we need to calculate
the likelihood of each data point under the model. The likelihood of a data point (xi, yi)under the
model is given by f(xi)⊖yi.
For the data point (2,7):
f(2) ⊖7=(a⊗2) ⊕b⊖7
For the data point (3,5):
f(3) ⊖5=(a⊗3) ⊕b⊖5
For the data point (4,3):
f(4) ⊖3=(a⊗4) ⊕b⊖3
b) The maximum likelihood estimation aims to find the values of aand bthat minimize the
difference between the model’s prediction and the observed data. Thus, we want to find aand b
that minimize the sum of squares of the differences:
Likelihood =X
i
[(a⊗xi)⊕b−yi]2
Expanding this expression and plugging in the data points, we get:
Likelihood = ((2a)⊕b−7)2+ ((3a)⊕b−5)2+ ((4a)⊕b−3)2
To minimize this expression, we differentiate with respect to aand b, set the derivatives to 0,
and solve for aand b.
c) Solving the system of equations, we find a= 2 and b= 3. Therefore, the tropical polynomial
that maximizes the likelihood of the data is f(x) = 2 ⊗x⊕3.
5 "POLYHEDRAL REALIZATIONS OF TROPICAL VARIETIES"
Problem 3. Consider the tropical variety Vdefined by the following sets of tropical polynomials:
f1(x, y) = max(3x+y, x + 2y), f2(x, y) = max(x+y, 2x−y)
a) Determine the critical points of the tropical variety V.
b) Find the polyhedral complex that realizes the tropical variety V.
Solution 3.
a) To determine the critical points of the tropical variety V, we need to find where the tropical
polynomials f1and f2have non-unique maximum values.
For f1(x, y) = max(3x+y, x + 2y), we set 3x+y=x+ 2yto find the critical points. Solving this
system of equations gives x=y. The corresponding value of f1at this critical point is f1(x, y) =
max(4x, 2x) = 4x.
For f2(x, y) = max(x+y, 2x−y), we set x+y= 2x−yto find the critical points. Solving this
system of equations gives y=x. The corresponding value of f2at this critical point is f2(x, y) =
max(2x, 2x) = 2x.
Thus, the critical points of the tropical variety Vare points where x=ywith maximum value 4x
under f1, and points where y=xwith maximum value 2xunder f2.
b) To find the polyhedral complex that realizes the tropical variety V, we construct the Newton
polytope of each tropical polynomial.
The Newton polytope of f1(x, y) = max(3x+y, x + 2y)is given by the convex hull of the points
(3,1) and (1,2), which forms a line segment in the xy-plane connecting these two points.
The Newton polytope of f2(x, y) = max(x+y, 2x−y)is given by the convex hull of the points
(1,1) and (2,−1), which forms a line segment in the xy-plane connecting these two points.
Next, we take the union of these two line segments to obtain the polyhedral complex that realizes
the tropical variety V. Therefore, the polyhedral complex corresponding to the tropical variety Vis
a union of two line segments in the xy-plane.
6 "OPTIMAL LINEAR PROGRAMMING IN TROPICAL CONVEX GEOMETRY"
Problem 9. Consider the following system of tropical linear inequalities in variables x,y, and
z:
2x⊕y⊕z≤3
3x⊕2y⊕z≤4
x⊕z≤2
Solve the system and find the optimal values of x,y, and zthat satisfy all the inequalities.
Solution 9. To solve the given system of tropical linear inequalities, we need to minimize the
right-hand side of each inequality and determine the optimal values of x,y, and zthat satisfy all
the inequalities.
a) For the first inequality, minimizing the right-hand side 3gives 2⊕0⊕1=2. Thus, the first
inequality becomes 2x⊕y⊕z≤2.
b) For the second inequality, minimizing the right-hand side 4gives 3⊕0⊕1=3. Thus, the
second inequality becomes 3x⊕2y⊕z≤3.
c) For the third inequality, minimizing the right-hand side 2gives 0⊕0=0. Thus, the third
inequality remains the same as x⊕z≤2.
Now, we have the simplified system of inequalities:
2x⊕y⊕z≤2
3x⊕2y⊕z≤3
x⊕z≤2
To find the optimal values of x,y, and z, we solve these inequalities simultaneously.
From the third inequality x⊕z≤2, we can obtain x≤2and z≤2.
Substitute z≤2into the first inequality 2x⊕y⊕z≤2, we get 2x⊕y≤2. Since x≤2and
2⊕0 = 2, this simplifies to 2=2x⊕y.
Similarly, substitute z≤2into the second inequality 3x⊕2y⊕z≤3, we have 3x⊕2y≤3.
Since x≤2and y≥0,3⊕0=3, we get that 3=3x⊕2y.
Solving 2=2x⊕yand 3=3x⊕2ysimultaneously gives x= 1,y= 0, and z= 2 as the optimal
values that satisfy all the inequalities.
Therefore, the optimal values for x,y, and zare x= 1,y= 0, and z= 2 in the given system.
I. Problem: Consider the tropical polynomial f(x, y) = max{3x+ 2y, x + 4y, 2x+ 2y}in R2.
a) Compute the minimum of f(x, y).
b) Find the points (x, y)where the minimum is attained.
II. Solution:
a) To find the minimum of f(x, y), we look for the minimum value among the three terms inside
the max function. Since the max function takes the maximum value, the minimum of a tropical
polynomial is the minimum of the coefficients.
Therefore, the minimum of f(x, y)is min{3,1,2}= 1.
b) To find the points (x, y)where the minimum is attained, we set each of the terms inside the
max function to the minimum value found in part (a) and solve for xand y.
When 3x+ 2y= 1, we get 3x+ 2y= 1 ⇒3x+ 2y= 1.
When x+ 4y= 1, we have x+ 4y= 1 ⇒x+ 4y= 1.
And when 2x+ 2y= 1,2x+ 2y= 1 ⇒2x+ 2y= 1.
Solving these systems of linear equations, we find that the minimum value of 1 is attained at
the points (x, y) = (1/5,2/5),(1/6,1/6),and (1/3,0).
7 ENUMERATIVE PROBLEMS IN TROPICAL INTERSECTION THEORY
Problem 4. Consider the following tropical hypersurfaces in R2:
H1:x⊕y= 1, H2: 2x⊕2y= 3, H3:x⊕2y= 2.
a) Find all tropical points of intersection of H1,H2, and H3.
b) Calculate the number of bounded regions in R2formed by these tropical hypersurfaces.
c) Determine the tropical intersection multiplicity of the tropical hypersurfaces H1,H2, and H3
at the tropical points of intersection.
Solution 4. a) To find the tropical points of intersection, we need to solve the system of tropical
equations x⊕y= 1,2x⊕2y= 3, and x⊕2y= 2. Solving each pair of equations gives:
x⊕y= 1 ⇒x= 1, y = 0,
2x⊕2y= 3 ⇒x= 2, y =1
2,
x⊕2y= 2 ⇒x= 2, y =1
2.
Hence, the tropical points of intersection are (1,0) and (2,1
2).
b) To determine the number of bounded regions in R2, we need to analyze the convex hull
formed by the tropical points of intersection. The points (1,0) and (2,1
2)lie on the same line x=
2−1
2y. Thus, the tropical hypersurfaces partition R2into 4 bounded regions.
c) Finally, to find the tropical intersection multiplicity, we count the number of times a bounded
region is bounded by the tropical hypersurfaces at the tropical points of intersection. At (1,0),
H1and H3intersect transversely, contributing intersection multiplicity 1 each. At (2,1
2), all three
hypersurfaces intersect, resulting in multiplicity 2.
Therefore, the tropical intersection multiplicity of the tropical hypersurfaces H1,H2and H3at
the tropical points of intersection is 1 + 1 + 2 = 4.
8 "WEIGHTED LAPLACIANS IN TROPICAL GEOMETRY"
Problem 6. Consider the weighted Laplacian of a graph Ggiven by the formula Lweighted(G) =
D−A, where Ais the adjacency matrix of Gand Dis the diagonal matrix containing the weighted
degrees of the vertices.
Let Gbe the graph shown below with edge weights indicated.
1 2
34
3
2 1
2
a) Calculate the adjacency matrix Afor graph G.
b) Calculate the weighted degrees of the vertices and write them in a diagonal matrix D.
c) Find the weighted Laplacian matrix Lweighted(G)for graph G.
Solution 6.
a) The adjacency matrix Afor graph Gis given by the following matrix:
A=
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
b) The weighted degrees of the vertices are the sum of the weights of the edges incident on
each vertex. Therefore, the diagonal matrix Dis:
D=diag(1 + 2 + 3,2+3+1,3+1+2,2 + 2) = diag(6,6,6,4) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
c) The weighted Laplacian matrix Lweighted(G)is then calculated as Lweighted(G) = D−A:
Lweighted(G) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
−
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
=
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
Therefore, the weighted Laplacian matrix for graph Gis
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
.
9 NUMERICAL PROBLEMS
Problem 1. Let v= (3,1,5) and w= (2,4,1) be two vectors in R3. Compute the tropical dot
product ⟨v, w⟩trop.
Problem 2. Consider the matrix M=
231
415
324
. Find the tropical rank of M.
Problem 3. Let A=2 4
5 3and B=3 1
2 4be two matrices. Compute the tropical eigen-
values of the matrix product AB.
10 SOLUTIONS
Solution 1. We compute the tropical dot product ⟨v, w⟩trop by taking the minimum of the entries in
each corresponding position:
⟨v, w⟩trop = (min(3,2),min(1,4),min(5,1)) = (2,1,1).
Therefore, ⟨v, w⟩trop = (2,1,1).
Solution 2. To find the tropical rank of the matrix M, we perform the tropical Gaussian elimi-
nation:
Step 1: Subtract the minimum entry in each row from all entries in that row.
0 1 0
3 0 4
1 0 3
Step 2: Subtract the minimum entry in each column from all entries in that column.
0 1 0
3 0 3
1 0 2
Step 3: Repeat Step 1 and Step 2 until it reaches a tropical zero matrix.
010
302
101
⇒
010
201
100
⇒
010
100
000
The rank of Min tropical algebra is 2.
Solution 3. To compute the tropical eigenvalues of the matrix product AB, we first calculate
AB:
AB =2 4
5 33 1
2 4=16 14
9 13.
The tropical eigenvalues of AB are obtained by taking the maximum entry in each correspond-
ing position of the matrix:
Max Eigenvalue 1 = max(16,14) = 16
Max Eigenvalue 2 = max(9,13) = 13.
Therefore, the tropical eigenvalues of AB are 16 and 13.
Therefore, the tropical convex hull of the algebraic curve Cis the set of four line segments that
intersect at the points (1,1),(−1,−1),(−1,1), and (1,−1).
This completes the solution to part (a).
3 "VERTEX DECOMPOSITIONS OF TROPICAL POLYTOPES"
Problem 10. Consider the tropical polytope Pin R3defined as the convex hull of the following
points: A(2,1,3),B(−1,4,2),C(0,2,5), and D(3,0,1). What is the vertex decomposition of P?
Solution 10. To find the vertex decomposition of P, we first need to determine the vertices
of the polytope. We start by considering all the possible tropical affine combinations of the given
points.
a) We need to find the vertices of P. Let’s set up the equation for the tropical affine combination:
Let V=λ1A+λ2B+λ3C+λ4Dbe a point on P.
The vertices are obtained by solving the following system of equations:
2λ1−λ2+λ3+ 3λ4= 0
λ1+ 4λ2+ 2λ3= 0
3λ1+ 2λ2+ 5λ3+λ4= 0
We can solve this system of equations to find the vertex points.
b) Solving the system of equations, we find that the vertices of Pare V1(1,1,2),V2(0,2,2),
V3(1,2,2), and V4(1,2,1).
c) The vertex decomposition of Pis given by the set of vertices {V1, V2, V3, V4}.
4 "MAXIMUM LIKELIHOOD ESTIMATION IN TROPICAL ALGEBRAIC COMBINATORICS"
Problem 1. Consider the following data points: {(2,7),(3,5),(4,3)}. We want to fit a tropical
polynomial f(x) = a⊗x⊕bto this data using the maximum likelihood estimation method. Find the
values of aand bthat maximize the likelihood of the data.
Solution 1.
a) To find the likelihood of the data given the model f(x) = a⊗x⊕b, we need to calculate
the likelihood of each data point under the model. The likelihood of a data point (xi, yi)under the
model is given by f(xi)⊖yi.
For the data point (2,7):
f(2) ⊖7=(a⊗2) ⊕b⊖7
For the data point (3,5):
f(3) ⊖5=(a⊗3) ⊕b⊖5
For the data point (4,3):
f(4) ⊖3=(a⊗4) ⊕b⊖3
b) The maximum likelihood estimation aims to find the values of aand bthat minimize the
difference between the model’s prediction and the observed data. Thus, we want to find aand b
that minimize the sum of squares of the differences:
Likelihood =X
i
[(a⊗xi)⊕b−yi]2
Expanding this expression and plugging in the data points, we get:
Likelihood = ((2a)⊕b−7)2+ ((3a)⊕b−5)2+ ((4a)⊕b−3)2
To minimize this expression, we differentiate with respect to aand b, set the derivatives to 0,
and solve for aand b.
c) Solving the system of equations, we find a= 2 and b= 3. Therefore, the tropical polynomial
that maximizes the likelihood of the data is f(x) = 2 ⊗x⊕3.
5 "POLYHEDRAL REALIZATIONS OF TROPICAL VARIETIES"
Problem 3. Consider the tropical variety Vdefined by the following sets of tropical polynomials:
f1(x, y) = max(3x+y, x + 2y), f2(x, y) = max(x+y, 2x−y)
a) Determine the critical points of the tropical variety V.
b) Find the polyhedral complex that realizes the tropical variety V.
Solution 3.
a) To determine the critical points of the tropical variety V, we need to find where the tropical
polynomials f1and f2have non-unique maximum values.
For f1(x, y) = max(3x+y, x + 2y), we set 3x+y=x+ 2yto find the critical points. Solving this
system of equations gives x=y. The corresponding value of f1at this critical point is f1(x, y) =
max(4x, 2x) = 4x.
For f2(x, y) = max(x+y, 2x−y), we set x+y= 2x−yto find the critical points. Solving this
system of equations gives y=x. The corresponding value of f2at this critical point is f2(x, y) =
max(2x, 2x) = 2x.
Thus, the critical points of the tropical variety Vare points where x=ywith maximum value 4x
under f1, and points where y=xwith maximum value 2xunder f2.
b) To find the polyhedral complex that realizes the tropical variety V, we construct the Newton
polytope of each tropical polynomial.
The Newton polytope of f1(x, y) = max(3x+y, x + 2y)is given by the convex hull of the points
(3,1) and (1,2), which forms a line segment in the xy-plane connecting these two points.
The Newton polytope of f2(x, y) = max(x+y, 2x−y)is given by the convex hull of the points
(1,1) and (2,−1), which forms a line segment in the xy-plane connecting these two points.
Next, we take the union of these two line segments to obtain the polyhedral complex that realizes
the tropical variety V. Therefore, the polyhedral complex corresponding to the tropical variety Vis
a union of two line segments in the xy-plane.
6 "OPTIMAL LINEAR PROGRAMMING IN TROPICAL CONVEX GEOMETRY"
Problem 9. Consider the following system of tropical linear inequalities in variables x,y, and
z:
2x⊕y⊕z≤3
3x⊕2y⊕z≤4
x⊕z≤2
Solve the system and find the optimal values of x,y, and zthat satisfy all the inequalities.
Solution 9. To solve the given system of tropical linear inequalities, we need to minimize the
right-hand side of each inequality and determine the optimal values of x,y, and zthat satisfy all
the inequalities.
a) For the first inequality, minimizing the right-hand side 3gives 2⊕0⊕1=2. Thus, the first
inequality becomes 2x⊕y⊕z≤2.
b) For the second inequality, minimizing the right-hand side 4gives 3⊕0⊕1=3. Thus, the
second inequality becomes 3x⊕2y⊕z≤3.
c) For the third inequality, minimizing the right-hand side 2gives 0⊕0=0. Thus, the third
inequality remains the same as x⊕z≤2.
Now, we have the simplified system of inequalities:
2x⊕y⊕z≤2
3x⊕2y⊕z≤3
x⊕z≤2
To find the optimal values of x,y, and z, we solve these inequalities simultaneously.
From the third inequality x⊕z≤2, we can obtain x≤2and z≤2.
Substitute z≤2into the first inequality 2x⊕y⊕z≤2, we get 2x⊕y≤2. Since x≤2and
2⊕0 = 2, this simplifies to 2=2x⊕y.
Similarly, substitute z≤2into the second inequality 3x⊕2y⊕z≤3, we have 3x⊕2y≤3.
Since x≤2and y≥0,3⊕0=3, we get that 3=3x⊕2y.
Solving 2=2x⊕yand 3=3x⊕2ysimultaneously gives x= 1,y= 0, and z= 2 as the optimal
values that satisfy all the inequalities.
Therefore, the optimal values for x,y, and zare x= 1,y= 0, and z= 2 in the given system.
I. Problem: Consider the tropical polynomial f(x, y) = max{3x+ 2y, x + 4y, 2x+ 2y}in R2.
a) Compute the minimum of f(x, y).
b) Find the points (x, y)where the minimum is attained.
II. Solution:
a) To find the minimum of f(x, y), we look for the minimum value among the three terms inside
the max function. Since the max function takes the maximum value, the minimum of a tropical
polynomial is the minimum of the coefficients.
Therefore, the minimum of f(x, y)is min{3,1,2}= 1.
b) To find the points (x, y)where the minimum is attained, we set each of the terms inside the
max function to the minimum value found in part (a) and solve for xand y.
When 3x+ 2y= 1, we get 3x+ 2y= 1 ⇒3x+ 2y= 1.
When x+ 4y= 1, we have x+ 4y= 1 ⇒x+ 4y= 1.
And when 2x+ 2y= 1,2x+ 2y= 1 ⇒2x+ 2y= 1.
Solving these systems of linear equations, we find that the minimum value of 1 is attained at
the points (x, y) = (1/5,2/5),(1/6,1/6),and (1/3,0).
7 ENUMERATIVE PROBLEMS IN TROPICAL INTERSECTION THEORY
Problem 4. Consider the following tropical hypersurfaces in R2:
H1:x⊕y= 1, H2: 2x⊕2y= 3, H3:x⊕2y= 2.
a) Find all tropical points of intersection of H1,H2, and H3.
b) Calculate the number of bounded regions in R2formed by these tropical hypersurfaces.
c) Determine the tropical intersection multiplicity of the tropical hypersurfaces H1,H2, and H3
at the tropical points of intersection.
Solution 4. a) To find the tropical points of intersection, we need to solve the system of tropical
equations x⊕y= 1,2x⊕2y= 3, and x⊕2y= 2. Solving each pair of equations gives:
x⊕y= 1 ⇒x= 1, y = 0,
2x⊕2y= 3 ⇒x= 2, y =1
2,
x⊕2y= 2 ⇒x= 2, y =1
2.
Hence, the tropical points of intersection are (1,0) and (2,1
2).
b) To determine the number of bounded regions in R2, we need to analyze the convex hull
formed by the tropical points of intersection. The points (1,0) and (2,1
2)lie on the same line x=
2−1
2y. Thus, the tropical hypersurfaces partition R2into 4 bounded regions.
c) Finally, to find the tropical intersection multiplicity, we count the number of times a bounded
region is bounded by the tropical hypersurfaces at the tropical points of intersection. At (1,0),
H1and H3intersect transversely, contributing intersection multiplicity 1 each. At (2,1
2), all three
hypersurfaces intersect, resulting in multiplicity 2.
Therefore, the tropical intersection multiplicity of the tropical hypersurfaces H1,H2and H3at
the tropical points of intersection is 1 + 1 + 2 = 4.
8 "WEIGHTED LAPLACIANS IN TROPICAL GEOMETRY"
Problem 6. Consider the weighted Laplacian of a graph Ggiven by the formula Lweighted(G) =
D−A, where Ais the adjacency matrix of Gand Dis the diagonal matrix containing the weighted
degrees of the vertices.
Let Gbe the graph shown below with edge weights indicated.
1 2
34
3
2 1
2
a) Calculate the adjacency matrix Afor graph G.
b) Calculate the weighted degrees of the vertices and write them in a diagonal matrix D.
c) Find the weighted Laplacian matrix Lweighted(G)for graph G.
Solution 6.
a) The adjacency matrix Afor graph Gis given by the following matrix:
A=
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
b) The weighted degrees of the vertices are the sum of the weights of the edges incident on
each vertex. Therefore, the diagonal matrix Dis:
D=diag(1 + 2 + 3,2+3+1,3+1+2,2 + 2) = diag(6,6,6,4) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
c) The weighted Laplacian matrix Lweighted(G)is then calculated as Lweighted(G) = D−A:
Lweighted(G) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
−
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
=
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
Therefore, the weighted Laplacian matrix for graph Gis
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
.
9 NUMERICAL PROBLEMS
Problem 1. Let v= (3,1,5) and w= (2,4,1) be two vectors in R3. Compute the tropical dot
product ⟨v, w⟩trop.
Problem 2. Consider the matrix M=
231
415
324
. Find the tropical rank of M.
Problem 3. Let A=2 4
5 3and B=3 1
2 4be two matrices. Compute the tropical eigen-
values of the matrix product AB.
10 SOLUTIONS
Solution 1. We compute the tropical dot product ⟨v, w⟩trop by taking the minimum of the entries in
each corresponding position:
⟨v, w⟩trop = (min(3,2),min(1,4),min(5,1)) = (2,1,1).
Therefore, ⟨v, w⟩trop = (2,1,1).
Solution 2. To find the tropical rank of the matrix M, we perform the tropical Gaussian elimi-
nation:
Step 1: Subtract the minimum entry in each row from all entries in that row.
0 1 0
3 0 4
1 0 3
Step 2: Subtract the minimum entry in each column from all entries in that column.
0 1 0
3 0 3
1 0 2
Step 3: Repeat Step 1 and Step 2 until it reaches a tropical zero matrix.
010
302
101
⇒
010
201
100
⇒
010
100
000
The rank of Min tropical algebra is 2.
Solution 3. To compute the tropical eigenvalues of the matrix product AB, we first calculate
AB:
AB =2 4
5 33 1
2 4=16 14
9 13.
The tropical eigenvalues of AB are obtained by taking the maximum entry in each correspond-
ing position of the matrix:
Max Eigenvalue 1 = max(16,14) = 16
Max Eigenvalue 2 = max(9,13) = 13.
Therefore, the tropical eigenvalues of AB are 16 and 13.
Therefore, the tropical convex hull of the algebraic curve Cis the set of four line segments that
intersect at the points (1,1),(−1,−1),(−1,1), and (1,−1).
This completes the solution to part (a).
3 "VERTEX DECOMPOSITIONS OF TROPICAL POLYTOPES"
Problem 10. Consider the tropical polytope Pin R3defined as the convex hull of the following
points: A(2,1,3),B(−1,4,2),C(0,2,5), and D(3,0,1). What is the vertex decomposition of P?
Solution 10. To find the vertex decomposition of P, we first need to determine the vertices
of the polytope. We start by considering all the possible tropical affine combinations of the given
points.
a) We need to find the vertices of P. Let’s set up the equation for the tropical affine combination:
Let V=λ1A+λ2B+λ3C+λ4Dbe a point on P.
The vertices are obtained by solving the following system of equations:
2λ1−λ2+λ3+ 3λ4= 0
λ1+ 4λ2+ 2λ3= 0
3λ1+ 2λ2+ 5λ3+λ4= 0
We can solve this system of equations to find the vertex points.
b) Solving the system of equations, we find that the vertices of Pare V1(1,1,2),V2(0,2,2),
V3(1,2,2), and V4(1,2,1).
c) The vertex decomposition of Pis given by the set of vertices {V1, V2, V3, V4}.
4 "MAXIMUM LIKELIHOOD ESTIMATION IN TROPICAL ALGEBRAIC COMBINATORICS"
Problem 1. Consider the following data points: {(2,7),(3,5),(4,3)}. We want to fit a tropical
polynomial f(x) = a⊗x⊕bto this data using the maximum likelihood estimation method. Find the
values of aand bthat maximize the likelihood of the data.
Solution 1.
a) To find the likelihood of the data given the model f(x) = a⊗x⊕b, we need to calculate
the likelihood of each data point under the model. The likelihood of a data point (xi, yi)under the
model is given by f(xi)⊖yi.
For the data point (2,7):
f(2) ⊖7=(a⊗2) ⊕b⊖7
For the data point (3,5):
f(3) ⊖5=(a⊗3) ⊕b⊖5
For the data point (4,3):
f(4) ⊖3=(a⊗4) ⊕b⊖3
b) The maximum likelihood estimation aims to find the values of aand bthat minimize the
difference between the model’s prediction and the observed data. Thus, we want to find aand b
that minimize the sum of squares of the differences:
Likelihood =X
i
[(a⊗xi)⊕b−yi]2
Expanding this expression and plugging in the data points, we get:
Likelihood = ((2a)⊕b−7)2+ ((3a)⊕b−5)2+ ((4a)⊕b−3)2
To minimize this expression, we differentiate with respect to aand b, set the derivatives to 0,
and solve for aand b.
c) Solving the system of equations, we find a= 2 and b= 3. Therefore, the tropical polynomial
that maximizes the likelihood of the data is f(x) = 2 ⊗x⊕3.
5 "POLYHEDRAL REALIZATIONS OF TROPICAL VARIETIES"
Problem 3. Consider the tropical variety Vdefined by the following sets of tropical polynomials:
f1(x, y) = max(3x+y, x + 2y), f2(x, y) = max(x+y, 2x−y)
a) Determine the critical points of the tropical variety V.
b) Find the polyhedral complex that realizes the tropical variety V.
Solution 3.
a) To determine the critical points of the tropical variety V, we need to find where the tropical
polynomials f1and f2have non-unique maximum values.
For f1(x, y) = max(3x+y, x + 2y), we set 3x+y=x+ 2yto find the critical points. Solving this
system of equations gives x=y. The corresponding value of f1at this critical point is f1(x, y) =
max(4x, 2x) = 4x.
For f2(x, y) = max(x+y, 2x−y), we set x+y= 2x−yto find the critical points. Solving this
system of equations gives y=x. The corresponding value of f2at this critical point is f2(x, y) =
max(2x, 2x) = 2x.
Thus, the critical points of the tropical variety Vare points where x=ywith maximum value 4x
under f1, and points where y=xwith maximum value 2xunder f2.
b) To find the polyhedral complex that realizes the tropical variety V, we construct the Newton
polytope of each tropical polynomial.
The Newton polytope of f1(x, y) = max(3x+y, x + 2y)is given by the convex hull of the points
(3,1) and (1,2), which forms a line segment in the xy-plane connecting these two points.
The Newton polytope of f2(x, y) = max(x+y, 2x−y)is given by the convex hull of the points
(1,1) and (2,−1), which forms a line segment in the xy-plane connecting these two points.
Next, we take the union of these two line segments to obtain the polyhedral complex that realizes
the tropical variety V. Therefore, the polyhedral complex corresponding to the tropical variety Vis
a union of two line segments in the xy-plane.
6 "OPTIMAL LINEAR PROGRAMMING IN TROPICAL CONVEX GEOMETRY"
Problem 9. Consider the following system of tropical linear inequalities in variables x,y, and
z:
2x⊕y⊕z≤3
3x⊕2y⊕z≤4
x⊕z≤2
Solve the system and find the optimal values of x,y, and zthat satisfy all the inequalities.
Solution 9. To solve the given system of tropical linear inequalities, we need to minimize the
right-hand side of each inequality and determine the optimal values of x,y, and zthat satisfy all
the inequalities.
a) For the first inequality, minimizing the right-hand side 3gives 2⊕0⊕1=2. Thus, the first
inequality becomes 2x⊕y⊕z≤2.
b) For the second inequality, minimizing the right-hand side 4gives 3⊕0⊕1=3. Thus, the
second inequality becomes 3x⊕2y⊕z≤3.
c) For the third inequality, minimizing the right-hand side 2gives 0⊕0=0. Thus, the third
inequality remains the same as x⊕z≤2.
Now, we have the simplified system of inequalities:
2x⊕y⊕z≤2
3x⊕2y⊕z≤3
x⊕z≤2
To find the optimal values of x,y, and z, we solve these inequalities simultaneously.
From the third inequality x⊕z≤2, we can obtain x≤2and z≤2.
Substitute z≤2into the first inequality 2x⊕y⊕z≤2, we get 2x⊕y≤2. Since x≤2and
2⊕0 = 2, this simplifies to 2=2x⊕y.
Similarly, substitute z≤2into the second inequality 3x⊕2y⊕z≤3, we have 3x⊕2y≤3.
Since x≤2and y≥0,3⊕0=3, we get that 3=3x⊕2y.
Solving 2=2x⊕yand 3=3x⊕2ysimultaneously gives x= 1,y= 0, and z= 2 as the optimal
values that satisfy all the inequalities.
Therefore, the optimal values for x,y, and zare x= 1,y= 0, and z= 2 in the given system.
I. Problem: Consider the tropical polynomial f(x, y) = max{3x+ 2y, x + 4y, 2x+ 2y}in R2.
a) Compute the minimum of f(x, y).
b) Find the points (x, y)where the minimum is attained.
II. Solution:
a) To find the minimum of f(x, y), we look for the minimum value among the three terms inside
the max function. Since the max function takes the maximum value, the minimum of a tropical
polynomial is the minimum of the coefficients.
Therefore, the minimum of f(x, y)is min{3,1,2}= 1.
b) To find the points (x, y)where the minimum is attained, we set each of the terms inside the
max function to the minimum value found in part (a) and solve for xand y.
When 3x+ 2y= 1, we get 3x+ 2y= 1 ⇒3x+ 2y= 1.
When x+ 4y= 1, we have x+ 4y= 1 ⇒x+ 4y= 1.
And when 2x+ 2y= 1,2x+ 2y= 1 ⇒2x+ 2y= 1.
Solving these systems of linear equations, we find that the minimum value of 1 is attained at
the points (x, y) = (1/5,2/5),(1/6,1/6),and (1/3,0).
7 ENUMERATIVE PROBLEMS IN TROPICAL INTERSECTION THEORY
Problem 4. Consider the following tropical hypersurfaces in R2:
H1:x⊕y= 1, H2: 2x⊕2y= 3, H3:x⊕2y= 2.
a) Find all tropical points of intersection of H1,H2, and H3.
b) Calculate the number of bounded regions in R2formed by these tropical hypersurfaces.
c) Determine the tropical intersection multiplicity of the tropical hypersurfaces H1,H2, and H3
at the tropical points of intersection.
Solution 4. a) To find the tropical points of intersection, we need to solve the system of tropical
equations x⊕y= 1,2x⊕2y= 3, and x⊕2y= 2. Solving each pair of equations gives:
x⊕y= 1 ⇒x= 1, y = 0,
2x⊕2y= 3 ⇒x= 2, y =1
2,
x⊕2y= 2 ⇒x= 2, y =1
2.
Hence, the tropical points of intersection are (1,0) and (2,1
2).
b) To determine the number of bounded regions in R2, we need to analyze the convex hull
formed by the tropical points of intersection. The points (1,0) and (2,1
2)lie on the same line x=
2−1
2y. Thus, the tropical hypersurfaces partition R2into 4 bounded regions.
c) Finally, to find the tropical intersection multiplicity, we count the number of times a bounded
region is bounded by the tropical hypersurfaces at the tropical points of intersection. At (1,0),
H1and H3intersect transversely, contributing intersection multiplicity 1 each. At (2,1
2), all three
hypersurfaces intersect, resulting in multiplicity 2.
Therefore, the tropical intersection multiplicity of the tropical hypersurfaces H1,H2and H3at
the tropical points of intersection is 1 + 1 + 2 = 4.
8 "WEIGHTED LAPLACIANS IN TROPICAL GEOMETRY"
Problem 6. Consider the weighted Laplacian of a graph Ggiven by the formula Lweighted(G) =
D−A, where Ais the adjacency matrix of Gand Dis the diagonal matrix containing the weighted
degrees of the vertices.
Let Gbe the graph shown below with edge weights indicated.
1 2
34
3
2 1
2
a) Calculate the adjacency matrix Afor graph G.
b) Calculate the weighted degrees of the vertices and write them in a diagonal matrix D.
c) Find the weighted Laplacian matrix Lweighted(G)for graph G.
Solution 6.
a) The adjacency matrix Afor graph Gis given by the following matrix:
A=
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
b) The weighted degrees of the vertices are the sum of the weights of the edges incident on
each vertex. Therefore, the diagonal matrix Dis:
D=diag(1 + 2 + 3,2+3+1,3+1+2,2 + 2) = diag(6,6,6,4) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
c) The weighted Laplacian matrix Lweighted(G)is then calculated as Lweighted(G) = D−A:
Lweighted(G) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
−
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
=
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
Therefore, the weighted Laplacian matrix for graph Gis
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
.
9 NUMERICAL PROBLEMS
Problem 1. Let v= (3,1,5) and w= (2,4,1) be two vectors in R3. Compute the tropical dot
product ⟨v, w⟩trop.
Problem 2. Consider the matrix M=
231
415
324
. Find the tropical rank of M.
Problem 3. Let A=2 4
5 3and B=3 1
2 4be two matrices. Compute the tropical eigen-
values of the matrix product AB.
10 SOLUTIONS
Solution 1. We compute the tropical dot product ⟨v, w⟩trop by taking the minimum of the entries in
each corresponding position:
⟨v, w⟩trop = (min(3,2),min(1,4),min(5,1)) = (2,1,1).
Therefore, ⟨v, w⟩trop = (2,1,1).
Solution 2. To find the tropical rank of the matrix M, we perform the tropical Gaussian elimi-
nation:
Step 1: Subtract the minimum entry in each row from all entries in that row.
0 1 0
3 0 4
1 0 3
Step 2: Subtract the minimum entry in each column from all entries in that column.
0 1 0
3 0 3
1 0 2
Step 3: Repeat Step 1 and Step 2 until it reaches a tropical zero matrix.
010
302
101
⇒
010
201
100
⇒
010
100
000
The rank of Min tropical algebra is 2.
Solution 3. To compute the tropical eigenvalues of the matrix product AB, we first calculate
AB:
AB =2 4
5 33 1
2 4=16 14
9 13.
The tropical eigenvalues of AB are obtained by taking the maximum entry in each correspond-
ing position of the matrix:
Max Eigenvalue 1 = max(16,14) = 16
Max Eigenvalue 2 = max(9,13) = 13.
Therefore, the tropical eigenvalues of AB are 16 and 13.
Therefore, the tropical convex hull of the algebraic curve Cis the set of four line segments that
intersect at the points (1,1),(−1,−1),(−1,1), and (1,−1).
This completes the solution to part (a).
3 "VERTEX DECOMPOSITIONS OF TROPICAL POLYTOPES"
Problem 10. Consider the tropical polytope Pin R3defined as the convex hull of the following
points: A(2,1,3),B(−1,4,2),C(0,2,5), and D(3,0,1). What is the vertex decomposition of P?
Solution 10. To find the vertex decomposition of P, we first need to determine the vertices
of the polytope. We start by considering all the possible tropical affine combinations of the given
points.
a) We need to find the vertices of P. Let’s set up the equation for the tropical affine combination:
Let V=λ1A+λ2B+λ3C+λ4Dbe a point on P.
The vertices are obtained by solving the following system of equations:
2λ1−λ2+λ3+ 3λ4= 0
λ1+ 4λ2+ 2λ3= 0
3λ1+ 2λ2+ 5λ3+λ4= 0
We can solve this system of equations to find the vertex points.
b) Solving the system of equations, we find that the vertices of Pare V1(1,1,2),V2(0,2,2),
V3(1,2,2), and V4(1,2,1).
c) The vertex decomposition of Pis given by the set of vertices {V1, V2, V3, V4}.
4 "MAXIMUM LIKELIHOOD ESTIMATION IN TROPICAL ALGEBRAIC COMBINATORICS"
Problem 1. Consider the following data points: {(2,7),(3,5),(4,3)}. We want to fit a tropical
polynomial f(x) = a⊗x⊕bto this data using the maximum likelihood estimation method. Find the
values of aand bthat maximize the likelihood of the data.
Solution 1.
a) To find the likelihood of the data given the model f(x) = a⊗x⊕b, we need to calculate
the likelihood of each data point under the model. The likelihood of a data point (xi, yi)under the
model is given by f(xi)⊖yi.
For the data point (2,7):
f(2) ⊖7=(a⊗2) ⊕b⊖7
For the data point (3,5):
f(3) ⊖5=(a⊗3) ⊕b⊖5
For the data point (4,3):
f(4) ⊖3=(a⊗4) ⊕b⊖3
b) The maximum likelihood estimation aims to find the values of aand bthat minimize the
difference between the model’s prediction and the observed data. Thus, we want to find aand b
that minimize the sum of squares of the differences:
Likelihood =X
i
[(a⊗xi)⊕b−yi]2
Expanding this expression and plugging in the data points, we get:
Likelihood = ((2a)⊕b−7)2+ ((3a)⊕b−5)2+ ((4a)⊕b−3)2
To minimize this expression, we differentiate with respect to aand b, set the derivatives to 0,
and solve for aand b.
c) Solving the system of equations, we find a= 2 and b= 3. Therefore, the tropical polynomial
that maximizes the likelihood of the data is f(x) = 2 ⊗x⊕3.
5 "POLYHEDRAL REALIZATIONS OF TROPICAL VARIETIES"
Problem 3. Consider the tropical variety Vdefined by the following sets of tropical polynomials:
f1(x, y) = max(3x+y, x + 2y), f2(x, y) = max(x+y, 2x−y)
a) Determine the critical points of the tropical variety V.
b) Find the polyhedral complex that realizes the tropical variety V.
Solution 3.
a) To determine the critical points of the tropical variety V, we need to find where the tropical
polynomials f1and f2have non-unique maximum values.
For f1(x, y) = max(3x+y, x + 2y), we set 3x+y=x+ 2yto find the critical points. Solving this
system of equations gives x=y. The corresponding value of f1at this critical point is f1(x, y) =
max(4x, 2x) = 4x.
For f2(x, y) = max(x+y, 2x−y), we set x+y= 2x−yto find the critical points. Solving this
system of equations gives y=x. The corresponding value of f2at this critical point is f2(x, y) =
max(2x, 2x) = 2x.
Thus, the critical points of the tropical variety Vare points where x=ywith maximum value 4x
under f1, and points where y=xwith maximum value 2xunder f2.
b) To find the polyhedral complex that realizes the tropical variety V, we construct the Newton
polytope of each tropical polynomial.
The Newton polytope of f1(x, y) = max(3x+y, x + 2y)is given by the convex hull of the points
(3,1) and (1,2), which forms a line segment in the xy-plane connecting these two points.
The Newton polytope of f2(x, y) = max(x+y, 2x−y)is given by the convex hull of the points
(1,1) and (2,−1), which forms a line segment in the xy-plane connecting these two points.
Next, we take the union of these two line segments to obtain the polyhedral complex that realizes
the tropical variety V. Therefore, the polyhedral complex corresponding to the tropical variety Vis
a union of two line segments in the xy-plane.
6 "OPTIMAL LINEAR PROGRAMMING IN TROPICAL CONVEX GEOMETRY"
Problem 9. Consider the following system of tropical linear inequalities in variables x,y, and
z:
2x⊕y⊕z≤3
3x⊕2y⊕z≤4
x⊕z≤2
Solve the system and find the optimal values of x,y, and zthat satisfy all the inequalities.
Solution 9. To solve the given system of tropical linear inequalities, we need to minimize the
right-hand side of each inequality and determine the optimal values of x,y, and zthat satisfy all
the inequalities.
a) For the first inequality, minimizing the right-hand side 3gives 2⊕0⊕1=2. Thus, the first
inequality becomes 2x⊕y⊕z≤2.
b) For the second inequality, minimizing the right-hand side 4gives 3⊕0⊕1=3. Thus, the
second inequality becomes 3x⊕2y⊕z≤3.
c) For the third inequality, minimizing the right-hand side 2gives 0⊕0=0. Thus, the third
inequality remains the same as x⊕z≤2.
Now, we have the simplified system of inequalities:
2x⊕y⊕z≤2
3x⊕2y⊕z≤3
x⊕z≤2
To find the optimal values of x,y, and z, we solve these inequalities simultaneously.
From the third inequality x⊕z≤2, we can obtain x≤2and z≤2.
Substitute z≤2into the first inequality 2x⊕y⊕z≤2, we get 2x⊕y≤2. Since x≤2and
2⊕0 = 2, this simplifies to 2=2x⊕y.
Similarly, substitute z≤2into the second inequality 3x⊕2y⊕z≤3, we have 3x⊕2y≤3.
Since x≤2and y≥0,3⊕0=3, we get that 3=3x⊕2y.
Solving 2=2x⊕yand 3=3x⊕2ysimultaneously gives x= 1,y= 0, and z= 2 as the optimal
values that satisfy all the inequalities.
Therefore, the optimal values for x,y, and zare x= 1,y= 0, and z= 2 in the given system.
I. Problem: Consider the tropical polynomial f(x, y) = max{3x+ 2y, x + 4y, 2x+ 2y}in R2.
a) Compute the minimum of f(x, y).
b) Find the points (x, y)where the minimum is attained.
II. Solution:
a) To find the minimum of f(x, y), we look for the minimum value among the three terms inside
the max function. Since the max function takes the maximum value, the minimum of a tropical
polynomial is the minimum of the coefficients.
Therefore, the minimum of f(x, y)is min{3,1,2}= 1.
b) To find the points (x, y)where the minimum is attained, we set each of the terms inside the
max function to the minimum value found in part (a) and solve for xand y.
When 3x+ 2y= 1, we get 3x+ 2y= 1 ⇒3x+ 2y= 1.
When x+ 4y= 1, we have x+ 4y= 1 ⇒x+ 4y= 1.
And when 2x+ 2y= 1,2x+ 2y= 1 ⇒2x+ 2y= 1.
Solving these systems of linear equations, we find that the minimum value of 1 is attained at
the points (x, y) = (1/5,2/5),(1/6,1/6),and (1/3,0).
7 ENUMERATIVE PROBLEMS IN TROPICAL INTERSECTION THEORY
Problem 4. Consider the following tropical hypersurfaces in R2:
H1:x⊕y= 1, H2: 2x⊕2y= 3, H3:x⊕2y= 2.
a) Find all tropical points of intersection of H1,H2, and H3.
b) Calculate the number of bounded regions in R2formed by these tropical hypersurfaces.
c) Determine the tropical intersection multiplicity of the tropical hypersurfaces H1,H2, and H3
at the tropical points of intersection.
Solution 4. a) To find the tropical points of intersection, we need to solve the system of tropical
equations x⊕y= 1,2x⊕2y= 3, and x⊕2y= 2. Solving each pair of equations gives:
x⊕y= 1 ⇒x= 1, y = 0,
2x⊕2y= 3 ⇒x= 2, y =1
2,
x⊕2y= 2 ⇒x= 2, y =1
2.
Hence, the tropical points of intersection are (1,0) and (2,1
2).
b) To determine the number of bounded regions in R2, we need to analyze the convex hull
formed by the tropical points of intersection. The points (1,0) and (2,1
2)lie on the same line x=
2−1
2y. Thus, the tropical hypersurfaces partition R2into 4 bounded regions.
c) Finally, to find the tropical intersection multiplicity, we count the number of times a bounded
region is bounded by the tropical hypersurfaces at the tropical points of intersection. At (1,0),
H1and H3intersect transversely, contributing intersection multiplicity 1 each. At (2,1
2), all three
hypersurfaces intersect, resulting in multiplicity 2.
Therefore, the tropical intersection multiplicity of the tropical hypersurfaces H1,H2and H3at
the tropical points of intersection is 1 + 1 + 2 = 4.
8 "WEIGHTED LAPLACIANS IN TROPICAL GEOMETRY"
Problem 6. Consider the weighted Laplacian of a graph Ggiven by the formula Lweighted(G) =
D−A, where Ais the adjacency matrix of Gand Dis the diagonal matrix containing the weighted
degrees of the vertices.
Let Gbe the graph shown below with edge weights indicated.
1 2
34
3
2 1
2
a) Calculate the adjacency matrix Afor graph G.
b) Calculate the weighted degrees of the vertices and write them in a diagonal matrix D.
c) Find the weighted Laplacian matrix Lweighted(G)for graph G.
Solution 6.
a) The adjacency matrix Afor graph Gis given by the following matrix:
A=
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
b) The weighted degrees of the vertices are the sum of the weights of the edges incident on
each vertex. Therefore, the diagonal matrix Dis:
D=diag(1 + 2 + 3,2+3+1,3+1+2,2 + 2) = diag(6,6,6,4) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
c) The weighted Laplacian matrix Lweighted(G)is then calculated as Lweighted(G) = D−A:
Lweighted(G) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
−
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
=
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
Therefore, the weighted Laplacian matrix for graph Gis
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
.
9 NUMERICAL PROBLEMS
Problem 1. Let v= (3,1,5) and w= (2,4,1) be two vectors in R3. Compute the tropical dot
product ⟨v, w⟩trop.
Problem 2. Consider the matrix M=
231
415
324
. Find the tropical rank of M.
Problem 3. Let A=2 4
5 3and B=3 1
2 4be two matrices. Compute the tropical eigen-
values of the matrix product AB.
10 SOLUTIONS
Solution 1. We compute the tropical dot product ⟨v, w⟩trop by taking the minimum of the entries in
each corresponding position:
⟨v, w⟩trop = (min(3,2),min(1,4),min(5,1)) = (2,1,1).
Therefore, ⟨v, w⟩trop = (2,1,1).
Solution 2. To find the tropical rank of the matrix M, we perform the tropical Gaussian elimi-
nation:
Step 1: Subtract the minimum entry in each row from all entries in that row.
0 1 0
3 0 4
1 0 3
Step 2: Subtract the minimum entry in each column from all entries in that column.
0 1 0
3 0 3
1 0 2
Step 3: Repeat Step 1 and Step 2 until it reaches a tropical zero matrix.
010
302
101
⇒
010
201
100
⇒
010
100
000
The rank of Min tropical algebra is 2.
Solution 3. To compute the tropical eigenvalues of the matrix product AB, we first calculate
AB:
AB =2 4
5 33 1
2 4=16 14
9 13.
The tropical eigenvalues of AB are obtained by taking the maximum entry in each correspond-
ing position of the matrix:
Max Eigenvalue 1 = max(16,14) = 16
Max Eigenvalue 2 = max(9,13) = 13.
Therefore, the tropical eigenvalues of AB are 16 and 13.
Therefore, the tropical convex hull of the algebraic curve Cis the set of four line segments that
intersect at the points (1,1),(−1,−1),(−1,1), and (1,−1).
This completes the solution to part (a).
3 "VERTEX DECOMPOSITIONS OF TROPICAL POLYTOPES"
Problem 10. Consider the tropical polytope Pin R3defined as the convex hull of the following
points: A(2,1,3),B(−1,4,2),C(0,2,5), and D(3,0,1). What is the vertex decomposition of P?
Solution 10. To find the vertex decomposition of P, we first need to determine the vertices
of the polytope. We start by considering all the possible tropical affine combinations of the given
points.
a) We need to find the vertices of P. Let’s set up the equation for the tropical affine combination:
Let V=λ1A+λ2B+λ3C+λ4Dbe a point on P.
The vertices are obtained by solving the following system of equations:
2λ1−λ2+λ3+ 3λ4= 0
λ1+ 4λ2+ 2λ3= 0
3λ1+ 2λ2+ 5λ3+λ4= 0
We can solve this system of equations to find the vertex points.
b) Solving the system of equations, we find that the vertices of Pare V1(1,1,2),V2(0,2,2),
V3(1,2,2), and V4(1,2,1).
c) The vertex decomposition of Pis given by the set of vertices {V1, V2, V3, V4}.
4 "MAXIMUM LIKELIHOOD ESTIMATION IN TROPICAL ALGEBRAIC COMBINATORICS"
Problem 1. Consider the following data points: {(2,7),(3,5),(4,3)}. We want to fit a tropical
polynomial f(x) = a⊗x⊕bto this data using the maximum likelihood estimation method. Find the
values of aand bthat maximize the likelihood of the data.
Solution 1.
a) To find the likelihood of the data given the model f(x) = a⊗x⊕b, we need to calculate
the likelihood of each data point under the model. The likelihood of a data point (xi, yi)under the
model is given by f(xi)⊖yi.
For the data point (2,7):
f(2) ⊖7=(a⊗2) ⊕b⊖7
For the data point (3,5):
f(3) ⊖5=(a⊗3) ⊕b⊖5
For the data point (4,3):
f(4) ⊖3=(a⊗4) ⊕b⊖3
b) The maximum likelihood estimation aims to find the values of aand bthat minimize the
difference between the model’s prediction and the observed data. Thus, we want to find aand b
that minimize the sum of squares of the differences:
Likelihood =X
i
[(a⊗xi)⊕b−yi]2
Expanding this expression and plugging in the data points, we get:
Likelihood = ((2a)⊕b−7)2+ ((3a)⊕b−5)2+ ((4a)⊕b−3)2
To minimize this expression, we differentiate with respect to aand b, set the derivatives to 0,
and solve for aand b.
c) Solving the system of equations, we find a= 2 and b= 3. Therefore, the tropical polynomial
that maximizes the likelihood of the data is f(x) = 2 ⊗x⊕3.
5 "POLYHEDRAL REALIZATIONS OF TROPICAL VARIETIES"
Problem 3. Consider the tropical variety Vdefined by the following sets of tropical polynomials:
f1(x, y) = max(3x+y, x + 2y), f2(x, y) = max(x+y, 2x−y)
a) Determine the critical points of the tropical variety V.
b) Find the polyhedral complex that realizes the tropical variety V.
Solution 3.
a) To determine the critical points of the tropical variety V, we need to find where the tropical
polynomials f1and f2have non-unique maximum values.
For f1(x, y) = max(3x+y, x + 2y), we set 3x+y=x+ 2yto find the critical points. Solving this
system of equations gives x=y. The corresponding value of f1at this critical point is f1(x, y) =
max(4x, 2x) = 4x.
For f2(x, y) = max(x+y, 2x−y), we set x+y= 2x−yto find the critical points. Solving this
system of equations gives y=x. The corresponding value of f2at this critical point is f2(x, y) =
max(2x, 2x) = 2x.
Thus, the critical points of the tropical variety Vare points where x=ywith maximum value 4x
under f1, and points where y=xwith maximum value 2xunder f2.
b) To find the polyhedral complex that realizes the tropical variety V, we construct the Newton
polytope of each tropical polynomial.
The Newton polytope of f1(x, y) = max(3x+y, x + 2y)is given by the convex hull of the points
(3,1) and (1,2), which forms a line segment in the xy-plane connecting these two points.
The Newton polytope of f2(x, y) = max(x+y, 2x−y)is given by the convex hull of the points
(1,1) and (2,−1), which forms a line segment in the xy-plane connecting these two points.
Next, we take the union of these two line segments to obtain the polyhedral complex that realizes
the tropical variety V. Therefore, the polyhedral complex corresponding to the tropical variety Vis
a union of two line segments in the xy-plane.
6 "OPTIMAL LINEAR PROGRAMMING IN TROPICAL CONVEX GEOMETRY"
Problem 9. Consider the following system of tropical linear inequalities in variables x,y, and
z:
2x⊕y⊕z≤3
3x⊕2y⊕z≤4
x⊕z≤2
Solve the system and find the optimal values of x,y, and zthat satisfy all the inequalities.
Solution 9. To solve the given system of tropical linear inequalities, we need to minimize the
right-hand side of each inequality and determine the optimal values of x,y, and zthat satisfy all
the inequalities.
a) For the first inequality, minimizing the right-hand side 3gives 2⊕0⊕1=2. Thus, the first
inequality becomes 2x⊕y⊕z≤2.
b) For the second inequality, minimizing the right-hand side 4gives 3⊕0⊕1=3. Thus, the
second inequality becomes 3x⊕2y⊕z≤3.
c) For the third inequality, minimizing the right-hand side 2gives 0⊕0=0. Thus, the third
inequality remains the same as x⊕z≤2.
Now, we have the simplified system of inequalities:
2x⊕y⊕z≤2
3x⊕2y⊕z≤3
x⊕z≤2
To find the optimal values of x,y, and z, we solve these inequalities simultaneously.
From the third inequality x⊕z≤2, we can obtain x≤2and z≤2.
Substitute z≤2into the first inequality 2x⊕y⊕z≤2, we get 2x⊕y≤2. Since x≤2and
2⊕0 = 2, this simplifies to 2=2x⊕y.
Similarly, substitute z≤2into the second inequality 3x⊕2y⊕z≤3, we have 3x⊕2y≤3.
Since x≤2and y≥0,3⊕0=3, we get that 3=3x⊕2y.
Solving 2=2x⊕yand 3=3x⊕2ysimultaneously gives x= 1,y= 0, and z= 2 as the optimal
values that satisfy all the inequalities.
Therefore, the optimal values for x,y, and zare x= 1,y= 0, and z= 2 in the given system.
I. Problem: Consider the tropical polynomial f(x, y) = max{3x+ 2y, x + 4y, 2x+ 2y}in R2.
a) Compute the minimum of f(x, y).
b) Find the points (x, y)where the minimum is attained.
II. Solution:
a) To find the minimum of f(x, y), we look for the minimum value among the three terms inside
the max function. Since the max function takes the maximum value, the minimum of a tropical
polynomial is the minimum of the coefficients.
Therefore, the minimum of f(x, y)is min{3,1,2}= 1.
b) To find the points (x, y)where the minimum is attained, we set each of the terms inside the
max function to the minimum value found in part (a) and solve for xand y.
When 3x+ 2y= 1, we get 3x+ 2y= 1 ⇒3x+ 2y= 1.
When x+ 4y= 1, we have x+ 4y= 1 ⇒x+ 4y= 1.
And when 2x+ 2y= 1,2x+ 2y= 1 ⇒2x+ 2y= 1.
Solving these systems of linear equations, we find that the minimum value of 1 is attained at
the points (x, y) = (1/5,2/5),(1/6,1/6),and (1/3,0).
7 ENUMERATIVE PROBLEMS IN TROPICAL INTERSECTION THEORY
Problem 4. Consider the following tropical hypersurfaces in R2:
H1:x⊕y= 1, H2: 2x⊕2y= 3, H3:x⊕2y= 2.
a) Find all tropical points of intersection of H1,H2, and H3.
b) Calculate the number of bounded regions in R2formed by these tropical hypersurfaces.
c) Determine the tropical intersection multiplicity of the tropical hypersurfaces H1,H2, and H3
at the tropical points of intersection.
Solution 4. a) To find the tropical points of intersection, we need to solve the system of tropical
equations x⊕y= 1,2x⊕2y= 3, and x⊕2y= 2. Solving each pair of equations gives:
x⊕y= 1 ⇒x= 1, y = 0,
2x⊕2y= 3 ⇒x= 2, y =1
2,
x⊕2y= 2 ⇒x= 2, y =1
2.
Hence, the tropical points of intersection are (1,0) and (2,1
2).
b) To determine the number of bounded regions in R2, we need to analyze the convex hull
formed by the tropical points of intersection. The points (1,0) and (2,1
2)lie on the same line x=
2−1
2y. Thus, the tropical hypersurfaces partition R2into 4 bounded regions.
c) Finally, to find the tropical intersection multiplicity, we count the number of times a bounded
region is bounded by the tropical hypersurfaces at the tropical points of intersection. At (1,0),
H1and H3intersect transversely, contributing intersection multiplicity 1 each. At (2,1
2), all three
hypersurfaces intersect, resulting in multiplicity 2.
Therefore, the tropical intersection multiplicity of the tropical hypersurfaces H1,H2and H3at
the tropical points of intersection is 1 + 1 + 2 = 4.
8 "WEIGHTED LAPLACIANS IN TROPICAL GEOMETRY"
Problem 6. Consider the weighted Laplacian of a graph Ggiven by the formula Lweighted(G) =
D−A, where Ais the adjacency matrix of Gand Dis the diagonal matrix containing the weighted
degrees of the vertices.
Let Gbe the graph shown below with edge weights indicated.
1 2
34
3
2 1
2
a) Calculate the adjacency matrix Afor graph G.
b) Calculate the weighted degrees of the vertices and write them in a diagonal matrix D.
c) Find the weighted Laplacian matrix Lweighted(G)for graph G.
Solution 6.
a) The adjacency matrix Afor graph Gis given by the following matrix:
A=
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
b) The weighted degrees of the vertices are the sum of the weights of the edges incident on
each vertex. Therefore, the diagonal matrix Dis:
D=diag(1 + 2 + 3,2+3+1,3+1+2,2 + 2) = diag(6,6,6,4) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
c) The weighted Laplacian matrix Lweighted(G)is then calculated as Lweighted(G) = D−A:
Lweighted(G) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
−
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
=
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
Therefore, the weighted Laplacian matrix for graph Gis
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
.
9 NUMERICAL PROBLEMS
Problem 1. Let v= (3,1,5) and w= (2,4,1) be two vectors in R3. Compute the tropical dot
product ⟨v, w⟩trop.
Problem 2. Consider the matrix M=
231
415
324
. Find the tropical rank of M.
Problem 3. Let A=2 4
5 3and B=3 1
2 4be two matrices. Compute the tropical eigen-
values of the matrix product AB.
10 SOLUTIONS
Solution 1. We compute the tropical dot product ⟨v, w⟩trop by taking the minimum of the entries in
each corresponding position:
⟨v, w⟩trop = (min(3,2),min(1,4),min(5,1)) = (2,1,1).
Therefore, ⟨v, w⟩trop = (2,1,1).
Solution 2. To find the tropical rank of the matrix M, we perform the tropical Gaussian elimi-
nation:
Step 1: Subtract the minimum entry in each row from all entries in that row.
0 1 0
3 0 4
1 0 3
Step 2: Subtract the minimum entry in each column from all entries in that column.
0 1 0
3 0 3
1 0 2
Step 3: Repeat Step 1 and Step 2 until it reaches a tropical zero matrix.
010
302
101
⇒
010
201
100
⇒
010
100
000
The rank of Min tropical algebra is 2.
Solution 3. To compute the tropical eigenvalues of the matrix product AB, we first calculate
AB:
AB =2 4
5 33 1
2 4=16 14
9 13.
The tropical eigenvalues of AB are obtained by taking the maximum entry in each correspond-
ing position of the matrix:
Max Eigenvalue 1 = max(16,14) = 16
Max Eigenvalue 2 = max(9,13) = 13.
Therefore, the tropical eigenvalues of AB are 16 and 13.
Therefore, the tropical convex hull of the algebraic curve Cis the set of four line segments that
intersect at the points (1,1),(−1,−1),(−1,1), and (1,−1).
This completes the solution to part (a).
3 "VERTEX DECOMPOSITIONS OF TROPICAL POLYTOPES"
Problem 10. Consider the tropical polytope Pin R3defined as the convex hull of the following
points: A(2,1,3),B(−1,4,2),C(0,2,5), and D(3,0,1). What is the vertex decomposition of P?
Solution 10. To find the vertex decomposition of P, we first need to determine the vertices
of the polytope. We start by considering all the possible tropical affine combinations of the given
points.
a) We need to find the vertices of P. Let’s set up the equation for the tropical affine combination:
Let V=λ1A+λ2B+λ3C+λ4Dbe a point on P.
The vertices are obtained by solving the following system of equations:
2λ1−λ2+λ3+ 3λ4= 0
λ1+ 4λ2+ 2λ3= 0
3λ1+ 2λ2+ 5λ3+λ4= 0
We can solve this system of equations to find the vertex points.
b) Solving the system of equations, we find that the vertices of Pare V1(1,1,2),V2(0,2,2),
V3(1,2,2), and V4(1,2,1).
c) The vertex decomposition of Pis given by the set of vertices {V1, V2, V3, V4}.
4 "MAXIMUM LIKELIHOOD ESTIMATION IN TROPICAL ALGEBRAIC COMBINATORICS"
Problem 1. Consider the following data points: {(2,7),(3,5),(4,3)}. We want to fit a tropical
polynomial f(x) = a⊗x⊕bto this data using the maximum likelihood estimation method. Find the
values of aand bthat maximize the likelihood of the data.
Solution 1.
a) To find the likelihood of the data given the model f(x) = a⊗x⊕b, we need to calculate
the likelihood of each data point under the model. The likelihood of a data point (xi, yi)under the
model is given by f(xi)⊖yi.
For the data point (2,7):
f(2) ⊖7=(a⊗2) ⊕b⊖7
For the data point (3,5):
f(3) ⊖5=(a⊗3) ⊕b⊖5
For the data point (4,3):
f(4) ⊖3=(a⊗4) ⊕b⊖3
b) The maximum likelihood estimation aims to find the values of aand bthat minimize the
difference between the model’s prediction and the observed data. Thus, we want to find aand b
that minimize the sum of squares of the differences:
Likelihood =X
i
[(a⊗xi)⊕b−yi]2
Expanding this expression and plugging in the data points, we get:
Likelihood = ((2a)⊕b−7)2+ ((3a)⊕b−5)2+ ((4a)⊕b−3)2
To minimize this expression, we differentiate with respect to aand b, set the derivatives to 0,
and solve for aand b.
c) Solving the system of equations, we find a= 2 and b= 3. Therefore, the tropical polynomial
that maximizes the likelihood of the data is f(x) = 2 ⊗x⊕3.
5 "POLYHEDRAL REALIZATIONS OF TROPICAL VARIETIES"
Problem 3. Consider the tropical variety Vdefined by the following sets of tropical polynomials:
f1(x, y) = max(3x+y, x + 2y), f2(x, y) = max(x+y, 2x−y)
a) Determine the critical points of the tropical variety V.
b) Find the polyhedral complex that realizes the tropical variety V.
Solution 3.
a) To determine the critical points of the tropical variety V, we need to find where the tropical
polynomials f1and f2have non-unique maximum values.
For f1(x, y) = max(3x+y, x + 2y), we set 3x+y=x+ 2yto find the critical points. Solving this
system of equations gives x=y. The corresponding value of f1at this critical point is f1(x, y) =
max(4x, 2x) = 4x.
For f2(x, y) = max(x+y, 2x−y), we set x+y= 2x−yto find the critical points. Solving this
system of equations gives y=x. The corresponding value of f2at this critical point is f2(x, y) =
max(2x, 2x) = 2x.
Thus, the critical points of the tropical variety Vare points where x=ywith maximum value 4x
under f1, and points where y=xwith maximum value 2xunder f2.
b) To find the polyhedral complex that realizes the tropical variety V, we construct the Newton
polytope of each tropical polynomial.
The Newton polytope of f1(x, y) = max(3x+y, x + 2y)is given by the convex hull of the points
(3,1) and (1,2), which forms a line segment in the xy-plane connecting these two points.
The Newton polytope of f2(x, y) = max(x+y, 2x−y)is given by the convex hull of the points
(1,1) and (2,−1), which forms a line segment in the xy-plane connecting these two points.
Next, we take the union of these two line segments to obtain the polyhedral complex that realizes
the tropical variety V. Therefore, the polyhedral complex corresponding to the tropical variety Vis
a union of two line segments in the xy-plane.
6 "OPTIMAL LINEAR PROGRAMMING IN TROPICAL CONVEX GEOMETRY"
Problem 9. Consider the following system of tropical linear inequalities in variables x,y, and
z:
2x⊕y⊕z≤3
3x⊕2y⊕z≤4
x⊕z≤2
Solve the system and find the optimal values of x,y, and zthat satisfy all the inequalities.
Solution 9. To solve the given system of tropical linear inequalities, we need to minimize the
right-hand side of each inequality and determine the optimal values of x,y, and zthat satisfy all
the inequalities.
a) For the first inequality, minimizing the right-hand side 3gives 2⊕0⊕1=2. Thus, the first
inequality becomes 2x⊕y⊕z≤2.
b) For the second inequality, minimizing the right-hand side 4gives 3⊕0⊕1=3. Thus, the
second inequality becomes 3x⊕2y⊕z≤3.
c) For the third inequality, minimizing the right-hand side 2gives 0⊕0=0. Thus, the third
inequality remains the same as x⊕z≤2.
Now, we have the simplified system of inequalities:
2x⊕y⊕z≤2
3x⊕2y⊕z≤3
x⊕z≤2
To find the optimal values of x,y, and z, we solve these inequalities simultaneously.
From the third inequality x⊕z≤2, we can obtain x≤2and z≤2.
Substitute z≤2into the first inequality 2x⊕y⊕z≤2, we get 2x⊕y≤2. Since x≤2and
2⊕0 = 2, this simplifies to 2=2x⊕y.
Similarly, substitute z≤2into the second inequality 3x⊕2y⊕z≤3, we have 3x⊕2y≤3.
Since x≤2and y≥0,3⊕0=3, we get that 3=3x⊕2y.
Solving 2=2x⊕yand 3=3x⊕2ysimultaneously gives x= 1,y= 0, and z= 2 as the optimal
values that satisfy all the inequalities.
Therefore, the optimal values for x,y, and zare x= 1,y= 0, and z= 2 in the given system.
I. Problem: Consider the tropical polynomial f(x, y) = max{3x+ 2y, x + 4y, 2x+ 2y}in R2.
a) Compute the minimum of f(x, y).
b) Find the points (x, y)where the minimum is attained.
II. Solution:
a) To find the minimum of f(x, y), we look for the minimum value among the three terms inside
the max function. Since the max function takes the maximum value, the minimum of a tropical
polynomial is the minimum of the coefficients.
Therefore, the minimum of f(x, y)is min{3,1,2}= 1.
b) To find the points (x, y)where the minimum is attained, we set each of the terms inside the
max function to the minimum value found in part (a) and solve for xand y.
When 3x+ 2y= 1, we get 3x+ 2y= 1 ⇒3x+ 2y= 1.
When x+ 4y= 1, we have x+ 4y= 1 ⇒x+ 4y= 1.
And when 2x+ 2y= 1,2x+ 2y= 1 ⇒2x+ 2y= 1.
Solving these systems of linear equations, we find that the minimum value of 1 is attained at
the points (x, y) = (1/5,2/5),(1/6,1/6),and (1/3,0).
7 ENUMERATIVE PROBLEMS IN TROPICAL INTERSECTION THEORY
Problem 4. Consider the following tropical hypersurfaces in R2:
H1:x⊕y= 1, H2: 2x⊕2y= 3, H3:x⊕2y= 2.
a) Find all tropical points of intersection of H1,H2, and H3.
b) Calculate the number of bounded regions in R2formed by these tropical hypersurfaces.
c) Determine the tropical intersection multiplicity of the tropical hypersurfaces H1,H2, and H3
at the tropical points of intersection.
Solution 4. a) To find the tropical points of intersection, we need to solve the system of tropical
equations x⊕y= 1,2x⊕2y= 3, and x⊕2y= 2. Solving each pair of equations gives:
x⊕y= 1 ⇒x= 1, y = 0,
2x⊕2y= 3 ⇒x= 2, y =1
2,
x⊕2y= 2 ⇒x= 2, y =1
2.
Hence, the tropical points of intersection are (1,0) and (2,1
2).
b) To determine the number of bounded regions in R2, we need to analyze the convex hull
formed by the tropical points of intersection. The points (1,0) and (2,1
2)lie on the same line x=
2−1
2y. Thus, the tropical hypersurfaces partition R2into 4 bounded regions.
c) Finally, to find the tropical intersection multiplicity, we count the number of times a bounded
region is bounded by the tropical hypersurfaces at the tropical points of intersection. At (1,0),
H1and H3intersect transversely, contributing intersection multiplicity 1 each. At (2,1
2), all three
hypersurfaces intersect, resulting in multiplicity 2.
Therefore, the tropical intersection multiplicity of the tropical hypersurfaces H1,H2and H3at
the tropical points of intersection is 1 + 1 + 2 = 4.
8 "WEIGHTED LAPLACIANS IN TROPICAL GEOMETRY"
Problem 6. Consider the weighted Laplacian of a graph Ggiven by the formula Lweighted(G) =
D−A, where Ais the adjacency matrix of Gand Dis the diagonal matrix containing the weighted
degrees of the vertices.
Let Gbe the graph shown below with edge weights indicated.
1 2
34
3
2 1
2
a) Calculate the adjacency matrix Afor graph G.
b) Calculate the weighted degrees of the vertices and write them in a diagonal matrix D.
c) Find the weighted Laplacian matrix Lweighted(G)for graph G.
Solution 6.
a) The adjacency matrix Afor graph Gis given by the following matrix:
A=
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
b) The weighted degrees of the vertices are the sum of the weights of the edges incident on
each vertex. Therefore, the diagonal matrix Dis:
D=diag(1 + 2 + 3,2+3+1,3+1+2,2 + 2) = diag(6,6,6,4) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
c) The weighted Laplacian matrix Lweighted(G)is then calculated as Lweighted(G) = D−A:
Lweighted(G) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
−
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
=
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
Therefore, the weighted Laplacian matrix for graph Gis
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
.
9 NUMERICAL PROBLEMS
Problem 1. Let v= (3,1,5) and w= (2,4,1) be two vectors in R3. Compute the tropical dot
product ⟨v, w⟩trop.
Problem 2. Consider the matrix M=
231
415
324
. Find the tropical rank of M.
Problem 3. Let A=2 4
5 3and B=3 1
2 4be two matrices. Compute the tropical eigen-
values of the matrix product AB.
10 SOLUTIONS
Solution 1. We compute the tropical dot product ⟨v, w⟩trop by taking the minimum of the entries in
each corresponding position:
⟨v, w⟩trop = (min(3,2),min(1,4),min(5,1)) = (2,1,1).
Therefore, ⟨v, w⟩trop = (2,1,1).
Solution 2. To find the tropical rank of the matrix M, we perform the tropical Gaussian elimi-
nation:
Step 1: Subtract the minimum entry in each row from all entries in that row.
0 1 0
3 0 4
1 0 3
Step 2: Subtract the minimum entry in each column from all entries in that column.
0 1 0
3 0 3
1 0 2
Step 3: Repeat Step 1 and Step 2 until it reaches a tropical zero matrix.
010
302
101
⇒
010
201
100
⇒
010
100
000
The rank of Min tropical algebra is 2.
Solution 3. To compute the tropical eigenvalues of the matrix product AB, we first calculate
AB:
AB =2 4
5 33 1
2 4=16 14
9 13.
The tropical eigenvalues of AB are obtained by taking the maximum entry in each correspond-
ing position of the matrix:
Max Eigenvalue 1 = max(16,14) = 16
Max Eigenvalue 2 = max(9,13) = 13.
Therefore, the tropical eigenvalues of AB are 16 and 13.
Therefore, the tropical convex hull of the algebraic curve Cis the set of four line segments that
intersect at the points (1,1),(−1,−1),(−1,1), and (1,−1).
This completes the solution to part (a).
3 "VERTEX DECOMPOSITIONS OF TROPICAL POLYTOPES"
Problem 10. Consider the tropical polytope Pin R3defined as the convex hull of the following
points: A(2,1,3),B(−1,4,2),C(0,2,5), and D(3,0,1). What is the vertex decomposition of P?
Solution 10. To find the vertex decomposition of P, we first need to determine the vertices
of the polytope. We start by considering all the possible tropical affine combinations of the given
points.
a) We need to find the vertices of P. Let’s set up the equation for the tropical affine combination:
Let V=λ1A+λ2B+λ3C+λ4Dbe a point on P.
The vertices are obtained by solving the following system of equations:
2λ1−λ2+λ3+ 3λ4= 0
λ1+ 4λ2+ 2λ3= 0
3λ1+ 2λ2+ 5λ3+λ4= 0
We can solve this system of equations to find the vertex points.
b) Solving the system of equations, we find that the vertices of Pare V1(1,1,2),V2(0,2,2),
V3(1,2,2), and V4(1,2,1).
c) The vertex decomposition of Pis given by the set of vertices {V1, V2, V3, V4}.
4 "MAXIMUM LIKELIHOOD ESTIMATION IN TROPICAL ALGEBRAIC COMBINATORICS"
Problem 1. Consider the following data points: {(2,7),(3,5),(4,3)}. We want to fit a tropical
polynomial f(x) = a⊗x⊕bto this data using the maximum likelihood estimation method. Find the
values of aand bthat maximize the likelihood of the data.
Solution 1.
a) To find the likelihood of the data given the model f(x) = a⊗x⊕b, we need to calculate
the likelihood of each data point under the model. The likelihood of a data point (xi, yi)under the
model is given by f(xi)⊖yi.
For the data point (2,7):
f(2) ⊖7=(a⊗2) ⊕b⊖7
For the data point (3,5):
f(3) ⊖5=(a⊗3) ⊕b⊖5
For the data point (4,3):
f(4) ⊖3=(a⊗4) ⊕b⊖3
b) The maximum likelihood estimation aims to find the values of aand bthat minimize the
difference between the model’s prediction and the observed data. Thus, we want to find aand b
that minimize the sum of squares of the differences:
Likelihood =X
i
[(a⊗xi)⊕b−yi]2
Expanding this expression and plugging in the data points, we get:
Likelihood = ((2a)⊕b−7)2+ ((3a)⊕b−5)2+ ((4a)⊕b−3)2
To minimize this expression, we differentiate with respect to aand b, set the derivatives to 0,
and solve for aand b.
c) Solving the system of equations, we find a= 2 and b= 3. Therefore, the tropical polynomial
that maximizes the likelihood of the data is f(x) = 2 ⊗x⊕3.
5 "POLYHEDRAL REALIZATIONS OF TROPICAL VARIETIES"
Problem 3. Consider the tropical variety Vdefined by the following sets of tropical polynomials:
f1(x, y) = max(3x+y, x + 2y), f2(x, y) = max(x+y, 2x−y)
a) Determine the critical points of the tropical variety V.
b) Find the polyhedral complex that realizes the tropical variety V.
Solution 3.
a) To determine the critical points of the tropical variety V, we need to find where the tropical
polynomials f1and f2have non-unique maximum values.
For f1(x, y) = max(3x+y, x + 2y), we set 3x+y=x+ 2yto find the critical points. Solving this
system of equations gives x=y. The corresponding value of f1at this critical point is f1(x, y) =
max(4x, 2x) = 4x.
For f2(x, y) = max(x+y, 2x−y), we set x+y= 2x−yto find the critical points. Solving this
system of equations gives y=x. The corresponding value of f2at this critical point is f2(x, y) =
max(2x, 2x) = 2x.
Thus, the critical points of the tropical variety Vare points where x=ywith maximum value 4x
under f1, and points where y=xwith maximum value 2xunder f2.
b) To find the polyhedral complex that realizes the tropical variety V, we construct the Newton
polytope of each tropical polynomial.
The Newton polytope of f1(x, y) = max(3x+y, x + 2y)is given by the convex hull of the points
(3,1) and (1,2), which forms a line segment in the xy-plane connecting these two points.
The Newton polytope of f2(x, y) = max(x+y, 2x−y)is given by the convex hull of the points
(1,1) and (2,−1), which forms a line segment in the xy-plane connecting these two points.
Next, we take the union of these two line segments to obtain the polyhedral complex that realizes
the tropical variety V. Therefore, the polyhedral complex corresponding to the tropical variety Vis
a union of two line segments in the xy-plane.
6 "OPTIMAL LINEAR PROGRAMMING IN TROPICAL CONVEX GEOMETRY"
Problem 9. Consider the following system of tropical linear inequalities in variables x,y, and
z:
2x⊕y⊕z≤3
3x⊕2y⊕z≤4
x⊕z≤2
Solve the system and find the optimal values of x,y, and zthat satisfy all the inequalities.
Solution 9. To solve the given system of tropical linear inequalities, we need to minimize the
right-hand side of each inequality and determine the optimal values of x,y, and zthat satisfy all
the inequalities.
a) For the first inequality, minimizing the right-hand side 3gives 2⊕0⊕1=2. Thus, the first
inequality becomes 2x⊕y⊕z≤2.
b) For the second inequality, minimizing the right-hand side 4gives 3⊕0⊕1=3. Thus, the
second inequality becomes 3x⊕2y⊕z≤3.
c) For the third inequality, minimizing the right-hand side 2gives 0⊕0=0. Thus, the third
inequality remains the same as x⊕z≤2.
Now, we have the simplified system of inequalities:
2x⊕y⊕z≤2
3x⊕2y⊕z≤3
x⊕z≤2
To find the optimal values of x,y, and z, we solve these inequalities simultaneously.
From the third inequality x⊕z≤2, we can obtain x≤2and z≤2.
Substitute z≤2into the first inequality 2x⊕y⊕z≤2, we get 2x⊕y≤2. Since x≤2and
2⊕0 = 2, this simplifies to 2=2x⊕y.
Similarly, substitute z≤2into the second inequality 3x⊕2y⊕z≤3, we have 3x⊕2y≤3.
Since x≤2and y≥0,3⊕0=3, we get that 3=3x⊕2y.
Solving 2=2x⊕yand 3=3x⊕2ysimultaneously gives x= 1,y= 0, and z= 2 as the optimal
values that satisfy all the inequalities.
Therefore, the optimal values for x,y, and zare x= 1,y= 0, and z= 2 in the given system.
I. Problem: Consider the tropical polynomial f(x, y) = max{3x+ 2y, x + 4y, 2x+ 2y}in R2.
a) Compute the minimum of f(x, y).
b) Find the points (x, y)where the minimum is attained.
II. Solution:
a) To find the minimum of f(x, y), we look for the minimum value among the three terms inside
the max function. Since the max function takes the maximum value, the minimum of a tropical
polynomial is the minimum of the coefficients.
Therefore, the minimum of f(x, y)is min{3,1,2}= 1.
b) To find the points (x, y)where the minimum is attained, we set each of the terms inside the
max function to the minimum value found in part (a) and solve for xand y.
When 3x+ 2y= 1, we get 3x+ 2y= 1 ⇒3x+ 2y= 1.
When x+ 4y= 1, we have x+ 4y= 1 ⇒x+ 4y= 1.
And when 2x+ 2y= 1,2x+ 2y= 1 ⇒2x+ 2y= 1.
Solving these systems of linear equations, we find that the minimum value of 1 is attained at
the points (x, y) = (1/5,2/5),(1/6,1/6),and (1/3,0).
7 ENUMERATIVE PROBLEMS IN TROPICAL INTERSECTION THEORY
Problem 4. Consider the following tropical hypersurfaces in R2:
H1:x⊕y= 1, H2: 2x⊕2y= 3, H3:x⊕2y= 2.
a) Find all tropical points of intersection of H1,H2, and H3.
b) Calculate the number of bounded regions in R2formed by these tropical hypersurfaces.
c) Determine the tropical intersection multiplicity of the tropical hypersurfaces H1,H2, and H3
at the tropical points of intersection.
Solution 4. a) To find the tropical points of intersection, we need to solve the system of tropical
equations x⊕y= 1,2x⊕2y= 3, and x⊕2y= 2. Solving each pair of equations gives:
x⊕y= 1 ⇒x= 1, y = 0,
2x⊕2y= 3 ⇒x= 2, y =1
2,
x⊕2y= 2 ⇒x= 2, y =1
2.
Hence, the tropical points of intersection are (1,0) and (2,1
2).
b) To determine the number of bounded regions in R2, we need to analyze the convex hull
formed by the tropical points of intersection. The points (1,0) and (2,1
2)lie on the same line x=
2−1
2y. Thus, the tropical hypersurfaces partition R2into 4 bounded regions.
c) Finally, to find the tropical intersection multiplicity, we count the number of times a bounded
region is bounded by the tropical hypersurfaces at the tropical points of intersection. At (1,0),
H1and H3intersect transversely, contributing intersection multiplicity 1 each. At (2,1
2), all three
hypersurfaces intersect, resulting in multiplicity 2.
Therefore, the tropical intersection multiplicity of the tropical hypersurfaces H1,H2and H3at
the tropical points of intersection is 1 + 1 + 2 = 4.
8 "WEIGHTED LAPLACIANS IN TROPICAL GEOMETRY"
Problem 6. Consider the weighted Laplacian of a graph Ggiven by the formula Lweighted(G) =
D−A, where Ais the adjacency matrix of Gand Dis the diagonal matrix containing the weighted
degrees of the vertices.
Let Gbe the graph shown below with edge weights indicated.
1 2
34
3
2 1
2
a) Calculate the adjacency matrix Afor graph G.
b) Calculate the weighted degrees of the vertices and write them in a diagonal matrix D.
c) Find the weighted Laplacian matrix Lweighted(G)for graph G.
Solution 6.
a) The adjacency matrix Afor graph Gis given by the following matrix:
A=
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
b) The weighted degrees of the vertices are the sum of the weights of the edges incident on
each vertex. Therefore, the diagonal matrix Dis:
D=diag(1 + 2 + 3,2+3+1,3+1+2,2 + 2) = diag(6,6,6,4) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
c) The weighted Laplacian matrix Lweighted(G)is then calculated as Lweighted(G) = D−A:
Lweighted(G) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
−
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
=
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
Therefore, the weighted Laplacian matrix for graph Gis
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
.
9 NUMERICAL PROBLEMS
Problem 1. Let v= (3,1,5) and w= (2,4,1) be two vectors in R3. Compute the tropical dot
product ⟨v, w⟩trop.
Problem 2. Consider the matrix M=
231
415
324
. Find the tropical rank of M.
Problem 3. Let A=2 4
5 3and B=3 1
2 4be two matrices. Compute the tropical eigen-
values of the matrix product AB.
10 SOLUTIONS
Solution 1. We compute the tropical dot product ⟨v, w⟩trop by taking the minimum of the entries in
each corresponding position:
⟨v, w⟩trop = (min(3,2),min(1,4),min(5,1)) = (2,1,1).
Therefore, ⟨v, w⟩trop = (2,1,1).
Solution 2. To find the tropical rank of the matrix M, we perform the tropical Gaussian elimi-
nation:
Step 1: Subtract the minimum entry in each row from all entries in that row.
0 1 0
3 0 4
1 0 3
Step 2: Subtract the minimum entry in each column from all entries in that column.
0 1 0
3 0 3
1 0 2
Step 3: Repeat Step 1 and Step 2 until it reaches a tropical zero matrix.
010
302
101
⇒
010
201
100
⇒
010
100
000
The rank of Min tropical algebra is 2.
Solution 3. To compute the tropical eigenvalues of the matrix product AB, we first calculate
AB:
AB =2 4
5 33 1
2 4=16 14
9 13.
The tropical eigenvalues of AB are obtained by taking the maximum entry in each correspond-
ing position of the matrix:
Max Eigenvalue 1 = max(16,14) = 16
Max Eigenvalue 2 = max(9,13) = 13.
Therefore, the tropical eigenvalues of AB are 16 and 13.
Therefore, the tropical convex hull of the algebraic curve Cis the set of four line segments that
intersect at the points (1,1),(−1,−1),(−1,1), and (1,−1).
This completes the solution to part (a).
3 "VERTEX DECOMPOSITIONS OF TROPICAL POLYTOPES"
Problem 10. Consider the tropical polytope Pin R3defined as the convex hull of the following
points: A(2,1,3),B(−1,4,2),C(0,2,5), and D(3,0,1). What is the vertex decomposition of P?
Solution 10. To find the vertex decomposition of P, we first need to determine the vertices
of the polytope. We start by considering all the possible tropical affine combinations of the given
points.
a) We need to find the vertices of P. Let’s set up the equation for the tropical affine combination:
Let V=λ1A+λ2B+λ3C+λ4Dbe a point on P.
The vertices are obtained by solving the following system of equations:
2λ1−λ2+λ3+ 3λ4= 0
λ1+ 4λ2+ 2λ3= 0
3λ1+ 2λ2+ 5λ3+λ4= 0
We can solve this system of equations to find the vertex points.
b) Solving the system of equations, we find that the vertices of Pare V1(1,1,2),V2(0,2,2),
V3(1,2,2), and V4(1,2,1).
c) The vertex decomposition of Pis given by the set of vertices {V1, V2, V3, V4}.
4 "MAXIMUM LIKELIHOOD ESTIMATION IN TROPICAL ALGEBRAIC COMBINATORICS"
Problem 1. Consider the following data points: {(2,7),(3,5),(4,3)}. We want to fit a tropical
polynomial f(x) = a⊗x⊕bto this data using the maximum likelihood estimation method. Find the
values of aand bthat maximize the likelihood of the data.
Solution 1.
a) To find the likelihood of the data given the model f(x) = a⊗x⊕b, we need to calculate
the likelihood of each data point under the model. The likelihood of a data point (xi, yi)under the
model is given by f(xi)⊖yi.
For the data point (2,7):
f(2) ⊖7=(a⊗2) ⊕b⊖7
For the data point (3,5):
f(3) ⊖5=(a⊗3) ⊕b⊖5
For the data point (4,3):
f(4) ⊖3=(a⊗4) ⊕b⊖3
b) The maximum likelihood estimation aims to find the values of aand bthat minimize the
difference between the model’s prediction and the observed data. Thus, we want to find aand b
that minimize the sum of squares of the differences:
Likelihood =X
i
[(a⊗xi)⊕b−yi]2
Expanding this expression and plugging in the data points, we get:
Likelihood = ((2a)⊕b−7)2+ ((3a)⊕b−5)2+ ((4a)⊕b−3)2
To minimize this expression, we differentiate with respect to aand b, set the derivatives to 0,
and solve for aand b.
c) Solving the system of equations, we find a= 2 and b= 3. Therefore, the tropical polynomial
that maximizes the likelihood of the data is f(x) = 2 ⊗x⊕3.
5 "POLYHEDRAL REALIZATIONS OF TROPICAL VARIETIES"
Problem 3. Consider the tropical variety Vdefined by the following sets of tropical polynomials:
f1(x, y) = max(3x+y, x + 2y), f2(x, y) = max(x+y, 2x−y)
a) Determine the critical points of the tropical variety V.
b) Find the polyhedral complex that realizes the tropical variety V.
Solution 3.
a) To determine the critical points of the tropical variety V, we need to find where the tropical
polynomials f1and f2have non-unique maximum values.
For f1(x, y) = max(3x+y, x + 2y), we set 3x+y=x+ 2yto find the critical points. Solving this
system of equations gives x=y. The corresponding value of f1at this critical point is f1(x, y) =
max(4x, 2x) = 4x.
For f2(x, y) = max(x+y, 2x−y), we set x+y= 2x−yto find the critical points. Solving this
system of equations gives y=x. The corresponding value of f2at this critical point is f2(x, y) =
max(2x, 2x) = 2x.
Thus, the critical points of the tropical variety Vare points where x=ywith maximum value 4x
under f1, and points where y=xwith maximum value 2xunder f2.
b) To find the polyhedral complex that realizes the tropical variety V, we construct the Newton
polytope of each tropical polynomial.
The Newton polytope of f1(x, y) = max(3x+y, x + 2y)is given by the convex hull of the points
(3,1) and (1,2), which forms a line segment in the xy-plane connecting these two points.
The Newton polytope of f2(x, y) = max(x+y, 2x−y)is given by the convex hull of the points
(1,1) and (2,−1), which forms a line segment in the xy-plane connecting these two points.
Next, we take the union of these two line segments to obtain the polyhedral complex that realizes
the tropical variety V. Therefore, the polyhedral complex corresponding to the tropical variety Vis
a union of two line segments in the xy-plane.
6 "OPTIMAL LINEAR PROGRAMMING IN TROPICAL CONVEX GEOMETRY"
Problem 9. Consider the following system of tropical linear inequalities in variables x,y, and
z:
2x⊕y⊕z≤3
3x⊕2y⊕z≤4
x⊕z≤2
Solve the system and find the optimal values of x,y, and zthat satisfy all the inequalities.
Solution 9. To solve the given system of tropical linear inequalities, we need to minimize the
right-hand side of each inequality and determine the optimal values of x,y, and zthat satisfy all
the inequalities.
a) For the first inequality, minimizing the right-hand side 3gives 2⊕0⊕1=2. Thus, the first
inequality becomes 2x⊕y⊕z≤2.
b) For the second inequality, minimizing the right-hand side 4gives 3⊕0⊕1=3. Thus, the
second inequality becomes 3x⊕2y⊕z≤3.
c) For the third inequality, minimizing the right-hand side 2gives 0⊕0=0. Thus, the third
inequality remains the same as x⊕z≤2.
Now, we have the simplified system of inequalities:
2x⊕y⊕z≤2
3x⊕2y⊕z≤3
x⊕z≤2
To find the optimal values of x,y, and z, we solve these inequalities simultaneously.
From the third inequality x⊕z≤2, we can obtain x≤2and z≤2.
Substitute z≤2into the first inequality 2x⊕y⊕z≤2, we get 2x⊕y≤2. Since x≤2and
2⊕0 = 2, this simplifies to 2=2x⊕y.
Similarly, substitute z≤2into the second inequality 3x⊕2y⊕z≤3, we have 3x⊕2y≤3.
Since x≤2and y≥0,3⊕0=3, we get that 3=3x⊕2y.
Solving 2=2x⊕yand 3=3x⊕2ysimultaneously gives x= 1,y= 0, and z= 2 as the optimal
values that satisfy all the inequalities.
Therefore, the optimal values for x,y, and zare x= 1,y= 0, and z= 2 in the given system.
I. Problem: Consider the tropical polynomial f(x, y) = max{3x+ 2y, x + 4y, 2x+ 2y}in R2.
a) Compute the minimum of f(x, y).
b) Find the points (x, y)where the minimum is attained.
II. Solution:
a) To find the minimum of f(x, y), we look for the minimum value among the three terms inside
the max function. Since the max function takes the maximum value, the minimum of a tropical
polynomial is the minimum of the coefficients.
Therefore, the minimum of f(x, y)is min{3,1,2}= 1.
b) To find the points (x, y)where the minimum is attained, we set each of the terms inside the
max function to the minimum value found in part (a) and solve for xand y.
When 3x+ 2y= 1, we get 3x+ 2y= 1 ⇒3x+ 2y= 1.
When x+ 4y= 1, we have x+ 4y= 1 ⇒x+ 4y= 1.
And when 2x+ 2y= 1,2x+ 2y= 1 ⇒2x+ 2y= 1.
Solving these systems of linear equations, we find that the minimum value of 1 is attained at
the points (x, y) = (1/5,2/5),(1/6,1/6),and (1/3,0).
7 ENUMERATIVE PROBLEMS IN TROPICAL INTERSECTION THEORY
Problem 4. Consider the following tropical hypersurfaces in R2:
H1:x⊕y= 1, H2: 2x⊕2y= 3, H3:x⊕2y= 2.
a) Find all tropical points of intersection of H1,H2, and H3.
b) Calculate the number of bounded regions in R2formed by these tropical hypersurfaces.
c) Determine the tropical intersection multiplicity of the tropical hypersurfaces H1,H2, and H3
at the tropical points of intersection.
Solution 4. a) To find the tropical points of intersection, we need to solve the system of tropical
equations x⊕y= 1,2x⊕2y= 3, and x⊕2y= 2. Solving each pair of equations gives:
x⊕y= 1 ⇒x= 1, y = 0,
2x⊕2y= 3 ⇒x= 2, y =1
2,
x⊕2y= 2 ⇒x= 2, y =1
2.
Hence, the tropical points of intersection are (1,0) and (2,1
2).
b) To determine the number of bounded regions in R2, we need to analyze the convex hull
formed by the tropical points of intersection. The points (1,0) and (2,1
2)lie on the same line x=
2−1
2y. Thus, the tropical hypersurfaces partition R2into 4 bounded regions.
c) Finally, to find the tropical intersection multiplicity, we count the number of times a bounded
region is bounded by the tropical hypersurfaces at the tropical points of intersection. At (1,0),
H1and H3intersect transversely, contributing intersection multiplicity 1 each. At (2,1
2), all three
hypersurfaces intersect, resulting in multiplicity 2.
Therefore, the tropical intersection multiplicity of the tropical hypersurfaces H1,H2and H3at
the tropical points of intersection is 1 + 1 + 2 = 4.
8 "WEIGHTED LAPLACIANS IN TROPICAL GEOMETRY"
Problem 6. Consider the weighted Laplacian of a graph Ggiven by the formula Lweighted(G) =
D−A, where Ais the adjacency matrix of Gand Dis the diagonal matrix containing the weighted
degrees of the vertices.
Let Gbe the graph shown below with edge weights indicated.
1 2
34
3
2 1
2
a) Calculate the adjacency matrix Afor graph G.
b) Calculate the weighted degrees of the vertices and write them in a diagonal matrix D.
c) Find the weighted Laplacian matrix Lweighted(G)for graph G.
Solution 6.
a) The adjacency matrix Afor graph Gis given by the following matrix:
A=
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
b) The weighted degrees of the vertices are the sum of the weights of the edges incident on
each vertex. Therefore, the diagonal matrix Dis:
D=diag(1 + 2 + 3,2+3+1,3+1+2,2 + 2) = diag(6,6,6,4) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
c) The weighted Laplacian matrix Lweighted(G)is then calculated as Lweighted(G) = D−A:
Lweighted(G) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
−
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
=
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
Therefore, the weighted Laplacian matrix for graph Gis
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
.
9 NUMERICAL PROBLEMS
Problem 1. Let v= (3,1,5) and w= (2,4,1) be two vectors in R3. Compute the tropical dot
product ⟨v, w⟩trop.
Problem 2. Consider the matrix M=
231
415
324
. Find the tropical rank of M.
Problem 3. Let A=2 4
5 3and B=3 1
2 4be two matrices. Compute the tropical eigen-
values of the matrix product AB.
10 SOLUTIONS
Solution 1. We compute the tropical dot product ⟨v, w⟩trop by taking the minimum of the entries in
each corresponding position:
⟨v, w⟩trop = (min(3,2),min(1,4),min(5,1)) = (2,1,1).
Therefore, ⟨v, w⟩trop = (2,1,1).
Solution 2. To find the tropical rank of the matrix M, we perform the tropical Gaussian elimi-
nation:
Step 1: Subtract the minimum entry in each row from all entries in that row.
0 1 0
3 0 4
1 0 3
Step 2: Subtract the minimum entry in each column from all entries in that column.
0 1 0
3 0 3
1 0 2
Step 3: Repeat Step 1 and Step 2 until it reaches a tropical zero matrix.
010
302
101
⇒
010
201
100
⇒
010
100
000
The rank of Min tropical algebra is 2.
Solution 3. To compute the tropical eigenvalues of the matrix product AB, we first calculate
AB:
AB =2 4
5 33 1
2 4=16 14
9 13.
The tropical eigenvalues of AB are obtained by taking the maximum entry in each correspond-
ing position of the matrix:
Max Eigenvalue 1 = max(16,14) = 16
Max Eigenvalue 2 = max(9,13) = 13.
Therefore, the tropical eigenvalues of AB are 16 and 13.
Therefore, the tropical convex hull of the algebraic curve Cis the set of four line segments that
intersect at the points (1,1),(−1,−1),(−1,1), and (1,−1).
This completes the solution to part (a).
3 "VERTEX DECOMPOSITIONS OF TROPICAL POLYTOPES"
Problem 10. Consider the tropical polytope Pin R3defined as the convex hull of the following
points: A(2,1,3),B(−1,4,2),C(0,2,5), and D(3,0,1). What is the vertex decomposition of P?
Solution 10. To find the vertex decomposition of P, we first need to determine the vertices
of the polytope. We start by considering all the possible tropical affine combinations of the given
points.
a) We need to find the vertices of P. Let’s set up the equation for the tropical affine combination:
Let V=λ1A+λ2B+λ3C+λ4Dbe a point on P.
The vertices are obtained by solving the following system of equations:
2λ1−λ2+λ3+ 3λ4= 0
λ1+ 4λ2+ 2λ3= 0
3λ1+ 2λ2+ 5λ3+λ4= 0
We can solve this system of equations to find the vertex points.
b) Solving the system of equations, we find that the vertices of Pare V1(1,1,2),V2(0,2,2),
V3(1,2,2), and V4(1,2,1).
c) The vertex decomposition of Pis given by the set of vertices {V1, V2, V3, V4}.
4 "MAXIMUM LIKELIHOOD ESTIMATION IN TROPICAL ALGEBRAIC COMBINATORICS"
Problem 1. Consider the following data points: {(2,7),(3,5),(4,3)}. We want to fit a tropical
polynomial f(x) = a⊗x⊕bto this data using the maximum likelihood estimation method. Find the
values of aand bthat maximize the likelihood of the data.
Solution 1.
a) To find the likelihood of the data given the model f(x) = a⊗x⊕b, we need to calculate
the likelihood of each data point under the model. The likelihood of a data point (xi, yi)under the
model is given by f(xi)⊖yi.
For the data point (2,7):
f(2) ⊖7=(a⊗2) ⊕b⊖7
For the data point (3,5):
f(3) ⊖5=(a⊗3) ⊕b⊖5
For the data point (4,3):
f(4) ⊖3=(a⊗4) ⊕b⊖3
b) The maximum likelihood estimation aims to find the values of aand bthat minimize the
difference between the model’s prediction and the observed data. Thus, we want to find aand b
that minimize the sum of squares of the differences:
Likelihood =X
i
[(a⊗xi)⊕b−yi]2
Expanding this expression and plugging in the data points, we get:
Likelihood = ((2a)⊕b−7)2+ ((3a)⊕b−5)2+ ((4a)⊕b−3)2
To minimize this expression, we differentiate with respect to aand b, set the derivatives to 0,
and solve for aand b.
c) Solving the system of equations, we find a= 2 and b= 3. Therefore, the tropical polynomial
that maximizes the likelihood of the data is f(x) = 2 ⊗x⊕3.
5 "POLYHEDRAL REALIZATIONS OF TROPICAL VARIETIES"
Problem 3. Consider the tropical variety Vdefined by the following sets of tropical polynomials:
f1(x, y) = max(3x+y, x + 2y), f2(x, y) = max(x+y, 2x−y)
a) Determine the critical points of the tropical variety V.
b) Find the polyhedral complex that realizes the tropical variety V.
Solution 3.
a) To determine the critical points of the tropical variety V, we need to find where the tropical
polynomials f1and f2have non-unique maximum values.
For f1(x, y) = max(3x+y, x + 2y), we set 3x+y=x+ 2yto find the critical points. Solving this
system of equations gives x=y. The corresponding value of f1at this critical point is f1(x, y) =
max(4x, 2x) = 4x.
For f2(x, y) = max(x+y, 2x−y), we set x+y= 2x−yto find the critical points. Solving this
system of equations gives y=x. The corresponding value of f2at this critical point is f2(x, y) =
max(2x, 2x) = 2x.
Thus, the critical points of the tropical variety Vare points where x=ywith maximum value 4x
under f1, and points where y=xwith maximum value 2xunder f2.
b) To find the polyhedral complex that realizes the tropical variety V, we construct the Newton
polytope of each tropical polynomial.
The Newton polytope of f1(x, y) = max(3x+y, x + 2y)is given by the convex hull of the points
(3,1) and (1,2), which forms a line segment in the xy-plane connecting these two points.
The Newton polytope of f2(x, y) = max(x+y, 2x−y)is given by the convex hull of the points
(1,1) and (2,−1), which forms a line segment in the xy-plane connecting these two points.
Next, we take the union of these two line segments to obtain the polyhedral complex that realizes
the tropical variety V. Therefore, the polyhedral complex corresponding to the tropical variety Vis
a union of two line segments in the xy-plane.
6 "OPTIMAL LINEAR PROGRAMMING IN TROPICAL CONVEX GEOMETRY"
Problem 9. Consider the following system of tropical linear inequalities in variables x,y, and
z:
2x⊕y⊕z≤3
3x⊕2y⊕z≤4
x⊕z≤2
Solve the system and find the optimal values of x,y, and zthat satisfy all the inequalities.
Solution 9. To solve the given system of tropical linear inequalities, we need to minimize the
right-hand side of each inequality and determine the optimal values of x,y, and zthat satisfy all
the inequalities.
a) For the first inequality, minimizing the right-hand side 3gives 2⊕0⊕1=2. Thus, the first
inequality becomes 2x⊕y⊕z≤2.
b) For the second inequality, minimizing the right-hand side 4gives 3⊕0⊕1=3. Thus, the
second inequality becomes 3x⊕2y⊕z≤3.
c) For the third inequality, minimizing the right-hand side 2gives 0⊕0=0. Thus, the third
inequality remains the same as x⊕z≤2.
Now, we have the simplified system of inequalities:
2x⊕y⊕z≤2
3x⊕2y⊕z≤3
x⊕z≤2
To find the optimal values of x,y, and z, we solve these inequalities simultaneously.
From the third inequality x⊕z≤2, we can obtain x≤2and z≤2.
Substitute z≤2into the first inequality 2x⊕y⊕z≤2, we get 2x⊕y≤2. Since x≤2and
2⊕0 = 2, this simplifies to 2=2x⊕y.
Similarly, substitute z≤2into the second inequality 3x⊕2y⊕z≤3, we have 3x⊕2y≤3.
Since x≤2and y≥0,3⊕0=3, we get that 3=3x⊕2y.
Solving 2=2x⊕yand 3=3x⊕2ysimultaneously gives x= 1,y= 0, and z= 2 as the optimal
values that satisfy all the inequalities.
Therefore, the optimal values for x,y, and zare x= 1,y= 0, and z= 2 in the given system.
I. Problem: Consider the tropical polynomial f(x, y) = max{3x+ 2y, x + 4y, 2x+ 2y}in R2.
a) Compute the minimum of f(x, y).
b) Find the points (x, y)where the minimum is attained.
II. Solution:
a) To find the minimum of f(x, y), we look for the minimum value among the three terms inside
the max function. Since the max function takes the maximum value, the minimum of a tropical
polynomial is the minimum of the coefficients.
Therefore, the minimum of f(x, y)is min{3,1,2}= 1.
b) To find the points (x, y)where the minimum is attained, we set each of the terms inside the
max function to the minimum value found in part (a) and solve for xand y.
When 3x+ 2y= 1, we get 3x+ 2y= 1 ⇒3x+ 2y= 1.
When x+ 4y= 1, we have x+ 4y= 1 ⇒x+ 4y= 1.
And when 2x+ 2y= 1,2x+ 2y= 1 ⇒2x+ 2y= 1.
Solving these systems of linear equations, we find that the minimum value of 1 is attained at
the points (x, y) = (1/5,2/5),(1/6,1/6),and (1/3,0).
7 ENUMERATIVE PROBLEMS IN TROPICAL INTERSECTION THEORY
Problem 4. Consider the following tropical hypersurfaces in R2:
H1:x⊕y= 1, H2: 2x⊕2y= 3, H3:x⊕2y= 2.
a) Find all tropical points of intersection of H1,H2, and H3.
b) Calculate the number of bounded regions in R2formed by these tropical hypersurfaces.
c) Determine the tropical intersection multiplicity of the tropical hypersurfaces H1,H2, and H3
at the tropical points of intersection.
Solution 4. a) To find the tropical points of intersection, we need to solve the system of tropical
equations x⊕y= 1,2x⊕2y= 3, and x⊕2y= 2. Solving each pair of equations gives:
x⊕y= 1 ⇒x= 1, y = 0,
2x⊕2y= 3 ⇒x= 2, y =1
2,
x⊕2y= 2 ⇒x= 2, y =1
2.
Hence, the tropical points of intersection are (1,0) and (2,1
2).
b) To determine the number of bounded regions in R2, we need to analyze the convex hull
formed by the tropical points of intersection. The points (1,0) and (2,1
2)lie on the same line x=
2−1
2y. Thus, the tropical hypersurfaces partition R2into 4 bounded regions.
c) Finally, to find the tropical intersection multiplicity, we count the number of times a bounded
region is bounded by the tropical hypersurfaces at the tropical points of intersection. At (1,0),
H1and H3intersect transversely, contributing intersection multiplicity 1 each. At (2,1
2), all three
hypersurfaces intersect, resulting in multiplicity 2.
Therefore, the tropical intersection multiplicity of the tropical hypersurfaces H1,H2and H3at
the tropical points of intersection is 1 + 1 + 2 = 4.
8 "WEIGHTED LAPLACIANS IN TROPICAL GEOMETRY"
Problem 6. Consider the weighted Laplacian of a graph Ggiven by the formula Lweighted(G) =
D−A, where Ais the adjacency matrix of Gand Dis the diagonal matrix containing the weighted
degrees of the vertices.
Let Gbe the graph shown below with edge weights indicated.
1 2
34
3
2 1
2
a) Calculate the adjacency matrix Afor graph G.
b) Calculate the weighted degrees of the vertices and write them in a diagonal matrix D.
c) Find the weighted Laplacian matrix Lweighted(G)for graph G.
Solution 6.
a) The adjacency matrix Afor graph Gis given by the following matrix:
A=
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
b) The weighted degrees of the vertices are the sum of the weights of the edges incident on
each vertex. Therefore, the diagonal matrix Dis:
D=diag(1 + 2 + 3,2+3+1,3+1+2,2 + 2) = diag(6,6,6,4) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
c) The weighted Laplacian matrix Lweighted(G)is then calculated as Lweighted(G) = D−A:
Lweighted(G) =
6 0 0 0
0 6 0 0
0 0 6 0
0 0 0 4
−
0 3 0 2
0 0 1 0
0 1 0 2
2 0 2 0
=
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
Therefore, the weighted Laplacian matrix for graph Gis
6−3 0 −2
0 6 −1 0
0−1 6 −2
−2 0 −2 4
.
9 NUMERICAL PROBLEMS
Problem 1. Let v= (3,1,5) and w= (2,4,1) be two vectors in R3. Compute the tropical dot
product ⟨v, w⟩trop.
Problem 2. Consider the matrix M=
231
415
324
. Find the tropical rank of M.
Problem 3. Let A=2 4
5 3and B=3 1
2 4be two matrices. Compute the tropical eigen-
values of the matrix product AB.
10 SOLUTIONS
Solution 1. We compute the tropical dot product ⟨v, w⟩trop by taking the minimum of the entries in
each corresponding position:
⟨v, w⟩trop = (min(3,2),min(1,4),min(5,1)) = (2,1,1).
Therefore, ⟨v, w⟩trop = (2,1,1).
Solution 2. To find the tropical rank of the matrix M, we perform the tropical Gaussian elimi-
nation:
Step 1: Subtract the minimum entry in each row from all entries in that row.
0 1 0
3 0 4
1 0 3
Step 2: Subtract the minimum entry in each column from all entries in that column.
0 1 0
3 0 3
1 0 2
Step 3: Repeat Step 1 and Step 2 until it reaches a tropical zero matrix.
010
302
101
⇒
010
201
100
⇒
010
100
000
The rank of Min tropical algebra is 2.
Solution 3. To compute the tropical eigenvalues of the matrix product AB, we first calculate
AB:
AB =2 4
5 33 1
2 4=16 14
9 13.
The tropical eigenvalues of AB are obtained by taking the maximum entry in each correspond-
ing position of the matrix:
Max Eigenvalue 1 = max(16,14) = 16
Max Eigenvalue 2 = max(9,13) = 13.
Therefore, the tropical eigenvalues of AB are 16 and 13.
