SPRING 2017: MTH 496-002 HOMEWORK 4 - DUE FRIDAY, 2/17 Solutions must be typed using LaTeX. If submitting by email, make sure that you send it to davisr@math.msu.edu 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 #3.3) Determine the generating function K (z) for the following cones: (a) K = {c1 (0, 1) + c2 (1, 0) | c1 , c2 0}; (b) K = {c1 (0, 1) + c2 (1, 1) | c1 , c2 0}; (c) K = {(3, 4) + c1 (0, 1) + c2 (2, 1) | c1 , c2 0}. 2. (Beck-Robins #3.8) Suppose X f (t)z t = t0 g(z) . (1 z)d+1 Prove that f (t) is a polynomial of degree exactly d if and only if g(z) is a polynomial of degree at most d and g(1) 6= 0. (Suggestion: for the \"forward\" direction, recall equation (2.2) in the book. For the \"backward\" direction, keep in mind Homework 3, #3.) 3. (Beck-Robins #3.11(a)) Compute the Ehrhart polynomial and Ehrhart series of P = conv{(0, 0, 0), (1, 0, 0), (0, 2, 0), (0, 0, 3)}. 4. (Beck-Robins, #3.16-ish) Given x0 , . . . , xd R, consider the matrix xd0 xd1 x0 1 0 xd xd1 x 1 1 1 1 V (x0 , . . . , xd ) = .. .. .. .. . . . . . . . . d1 d xd xd xd 1 Verify that det V (x0 , x1 , x2 ) = (x0 x1 )(x0 x2 )(x1 x2 ). 5. (Beck-Robins #3.18) Let Rh be the tetrahedron Rh = conv{(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, h)} where h is some positive integer. Prove that the vertices are the only lattice points of Rh . 6. Let SLn (Z) denote the set of nn matrices A with integer entries such that det A = 1. This set forms a group under multiplication. In particular, if A SLn (Z), then A is invertible and A1 SLn (Z). 1 2 SPRING 2017: MTH 496-002 HOMEWORK 4 - DUE FRIDAY, 2/17 Given A SLn (Z), define fA : Rn Rn by fA (x) = Ax. (a) For a fixed A SLn (Z), show that fA is a bijection. (b) Prove that if P Rn is a polytope and A SLn (Z), then fA (P ) is also a polytope. (c) We say that polytopes P and Q are unimodularly equivalent if there exists some A SLn (Z) such that Q = fA (P ). Prove that if P, Q Rn are unimodularly equivalent polytopes, then LP (t) = LQ (t). \f\f\f\f\f\f\f\f