Linear recursions for integer point transforms

Abstract

We consider the integer point transform σPn ] of a polytope P ⊂ R n . We show that if P is a lattice polytope then for any polytope Q the sequence {σ kP +Q (x)} k≥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).