MTH 400level class homework

hw03.tex

\documentclass[12pt]{amsart} \usepackage{amsmath,amssymb,amsthm} \usepackage[margin=1in]{geometry} \DeclareMathOperator{\BiPyr}{BiPyr} \DeclareMathOperator{\Ehr}{Ehr} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\PP}{\mathbb{P}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\RR}{\mathbb{R}} \newcommand{\ZZ}{\mathbb{Z}} \renewcommand{\theenumi}{\arabic{enumi}} \renewcommand\labelenumi{\theenumi.} %%%%%%%%%% %% Title and author %%%%%%%%%% \title{Spring 2017: MTH 496--002 \\ Homework 3 -- due Monday, 2/6} \author{YOUR NAME HERE} %% end of preamble %% beginning of document \begin{document} \maketitle 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. {\bf 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. \\ \begin{enumerate} %%%%%%% %%%%%%% % Question 1 %%%%%%% %%%%%%% \item (Beck-Robins \#2.1/2.3(a)) Fix positive integers $a,b,c,d$. \begin{enumerate} \item Suppose $\gcd(a,b) = \gcd(c,d) = 1$ and $a/b < c/d$. Let $P \subseteq \RR$ be the interval $[\frac{a}{b},\frac{c}{d}]$. Compute $L_P(t)$ and $L_{P^{\circ}}(t)$. \item Let $T$ be the triangle with vertices $(0,0),(a,0)$, and $(0,b)$. Use Homework 2 \#3 and an appropriate theorem to compute $L_T(t)$ and $\Ehr_T(z)$. \\ \end{enumerate} \begin{proof} Type your proof here! \end{proof} %%%%%%% %%%%%%% % Question 2 %%%%%%% %%%%%%% \item (Beck-Robins \#2.4) Let $P \subseteq \RR^m$ and $Q \subseteq \RR^n$ be any polytopes. \begin{enumerate} \item Prove that \[ \#((P \times Q) \cap \ZZ^{m+n}) = \#(P \cap \ZZ^m)\cdot \#(Q \cap \ZZ^n). \] \item Use part (a) to conclude that $L_{P \times Q}(t) = L_P(t)L_Q(t)$. \\ \end{enumerate} \begin{proof} Type your proof here! \end{proof} %%%%%%% %%%%%%% % Question 3 %%%%%%% %%%%%%% \item (Beck-Robins \#2.9) Let $d$ be a nonnegative integer. \begin{enumerate} \item Prove that \[ \frac{z^d}{(1-z)^{d+1}} = \sum_{k \geq 0} \binom{k}{d}z^k. \] \item Use part (a) to conclude that \[ \frac{1}{(1-z)^{d+1}} = \sum_{k \geq 0} \binom{d+k}{d}z^k. \] \end{enumerate} \begin{proof} Type your proof here! \end{proof} %%%%%%% %%%%%%% % Question 4 %%%%%%% %%%%%%% \item (Beck-Robins \#2.10) For $t,k \in \ZZ$ and $d \in \ZZ_{>0}$, prove that \[ (-1)^d\binom{-t+k}{d} = \binom{t+d-1-k}{d}. \] \begin{proof} Type your proof here! \end{proof} %%%%%%% %%%%%%% % Question 5 %%%%%%% %%%%%%% \item (Beck-Robins \#2.23) Prove that if $Q$ contains the origin, then \[ \Ehr_{\BiPyr(Q)}(z) = \frac{1+z}{1-z}\Ehr_Q(z). \] (Suggestion: start out by trying to recreate the proof of Theorem 2.4.) \\ \begin{proof} Type your proof here! \end{proof} %%%%%%% %%%%%%% % Question 6 %%%%%%% %%%%%%% \item (Beck-Robins \#2.26) Let $P$ be the self-intersecting polygon defined by the line segments \[ \begin{array}{c} [(0,0),(4,2)] \\ \relax [(4,2),(4,0)] \\ \relax [(4,0),(0,2)] \\ \relax [(0,2),(0,0)] \end{array} \] Show that Pick's theorem does not hold for $P$. \\ \begin{proof} Type your proof here! \end{proof} \end{enumerate} \end{document}