MTH 400level class homework

SPRING 2017: MTH 496–002 HOMEWORK 3 – DUE MONDAY, 2/6

Solutions must be typed using LaTeX. If submitting by email, make sure that you send it to [email protected] by 9:10 am on the due date. You will receive a confirmation to let you know that it has been received on time. Make absolutely sure to attach your homework file! If you submit a paper copy, make sure to staple it when submitting. Remember to provide justification for all responses unless otherwise noted.

1. (Beck-Robins #2.1/2.3(a)) Fix positive integers a, b, c, d. (a) Suppose gcd(a, b) = gcd(c, d) = 1 and a/b < c/d. Let P ⊆ R be the interval

[ a b , c d ]. Compute LP (t) and LP ◦ (t).

(b) Let T be the triangle with vertices (0, 0), (a, 0), and (0, b). Use Homework 2 #3 and an appropriate theorem to compute LT (t) and EhrT (z).

2. (Beck-Robins #2.4) Let P ⊆ Rm and Q ⊆ Rn be any polytopes. (a) Prove that

#((P ×Q) ∩Zm+n) = #(P ∩Zm) · #(Q∩Zn).

(b) Use part (a) to conclude that LP×Q(t) = LP (t)LQ(t).

3. (Beck-Robins #2.9) Let d be a nonnegative integer. (a) Prove that

zd

(1 −z)d+1 = ∑ k≥0

( k

d

) zk.

(b) Use part (a) to conclude that

1

(1 −z)d+1 = ∑ k≥0

( d + k

d

) zk.

4. (Beck-Robins #2.10) For t, k ∈ Z and d ∈ Z>0, prove that

(−1)d ( −t + k

d

) =

( t + d− 1 −k

d

) .

5. (Beck-Robins #2.23) Prove that if Q contains the origin, then

EhrBiPyr(Q)(z) = 1 + z

1 −z EhrQ(z).

(Suggestion: start out by trying to recreate the proof of Theorem 2.4.)

1

2 SPRING 2017: MTH 496–002 HOMEWORK 3 – DUE MONDAY, 2/6

6. (Beck-Robins #2.26) Let P be the self-intersecting polygon defined by the line seg- ments

[(0, 0), (4, 2)] [(4, 2), (4, 0)] [(4, 0), (0, 2)] [(0, 2), (0, 0)]

Show that Pick’s theorem does not hold for P .