SPRING 2017: MTH 496-002 HOMEWORK 3 - DUE MONDAY, 2/6 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 [ ab , dc ]. 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). Proof. Type your proof here! \u0003 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). Proof. Type your proof here! \u0003 3. (Beck-Robins #2.9) Let d be a nonnegative integer. (a) Prove that X \u0012k \u0013 zd = zk . d (1 z)d+1 k0 (b) Use part (a) to conclude that X \u0012d + k \u0013 1 = zk . d (1 z)d+1 k0 Proof. Type your proof here! \u0003 4. (Beck-Robins #2.10) For t, k Z and d Z>0 , prove that \u0012 \u0013 \u0012 \u0013 t+d1k d t + k (1) = . d d Proof. Type your proof here! \u0003 5. (Beck-Robins #2.23) Prove that if Q contains the origin, then 1+z EhrQ (z). 1z (Suggestion: start out by trying to recreate the proof of Theorem 2.4.) EhrBiPyr(Q) (z) = Proof. Type your proof here! \u0003 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 segments [(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 . Proof. Type your proof here! \u0003