1. [BeckRobins, #4.5[a,c)) Recall from Homework 2, #Hc] that Zzt+sz= 12'\" till as rational functions even though the left side involves an innite sun] of n1onon1ials with positive coefcients, cancellation occurs after rewriting each large suInnlancl as a rational function and simplifying. 1We'll prove that this happens more generally. Let Q : E. -} 1: he an}:r polynomial. and dene age} = 2 was and age) = Z Git)?- tgc mo [a] 'We know that REE\") evaluates to a rational function. Prove that Ra] also evaluates to a rational function. [h] Prove that1 as rational functions, Rah] + Has} 2 . 2. [BeckRobins, #4ish) Suppose P is a rational dclimensional polvtope that satises [a] Lpuft} = Lp[t In], and [h] LFE1J= Lpn{2) = - - . = Lpni: 1) = [1' for some integer Ll Prove that at... (i) = [1}d+lzf Ehrpr}. 3. [BeckRobins #4.?) Suppose P is a lattice [rtdimensional polytope with Ehrhart series 313:\" + h;_lz\"' + - - - + his + 1 EhIF[z} = {1 _ z)d+1 Prove that h; = Lpa}. c1. [BeckRobins #43} Suppose P is a lattice Isldimensional polytope. Prove that (tf+1]P contains an interior lattice point