Linear recursions for integer point transforms

Abstract: We consider the integer point transform $\sigma _P (\mathbf{x}) = \sum _{\mathbf{m} \in P\cap \mathbb{Z}^n} \mathbf{x}^\mathbf{m} \in \mathbb C [x_1^{\pm 1},\ldots, x_n^{\pm 1}]$ of a polytope $P\subset \mathbb{R}^n$. We show that if $P$ is a lattice polytope then for any polytope $Q$ the sequence $\lbrace \sigma _{kP+Q}(\mathbf{x})\rbrace _{k\geq 0}$ 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).