Linear recursions for integer point transforms

Abstract

We consider the integer point transform ${\sigma }_P(\mathbf{x})={\sum }_{\mathbf{m}\in P\cap {\mathbb{Z}}^n}{\mathbf{x}}^{\mathbf{m}}\in \mathbb{C}[x_1^{\pm 1},\dots ,x_n^{\pm 1}]$ of a polytope $P\subset {\mathbb{R}}^n$. We show that if $P$ is a lattice polytope then for any polytope $Q$ the sequence $\{{\sigma }_{kP+Q}(\mathbf{x}){\}}_{k\ge 0}$ satisfies a multivariate linear recursion that only depends on the vertices of $P$. We recover Brion's Theorem and by applying our results to Schur polynomials we disprove a conjecture of Alexandersson (2014).

8 pages, 2 figures; to appear in "Interactions with Lattice Polytopes; Magdeburg, Germany, September 2017; Springer Proceedings in Mathematics and Statistics"