Lattice points in vector-dilated polytopes

TL;DR

This paper extends the understanding of lattice point counts in polytopes defined by integer matrices, showing that for rational dilations, the count behaves as a piecewise polynomial function, generalizing known quasi-polynomial results.

Contribution

It generalizes the quasi-polynomial lattice point counting to rational vectors and reveals the polynomial nature of coefficients in this broader context.

Findings

Number of lattice points is a quasi-polynomial for integral vectors.

Coefficients are piecewise-defined polynomials for rational vectors.

Extension of McMullen's theorem to rational dilates of polytopes.

Abstract

For $A\in\mathbb{Z}^{m\times n}$ we investigate the behaviour of the number of lattice points in $P_A(b)=\{x\in\mathbb{R}^n:Ax\leq b\}$, depending on the varying vector $b$. It is known that this number, restricted to a cone of constant combinatorial type of $P_A(b)$, is a quasi-polynomial function if b is an integral vector. We extend this result to rational vectors $b$ and show that the coefficients themselves are piecewise-defined polynomials. To this end, we use a theorem of McMullen on lattice points in Minkowski-sums of rational dilates of rational polytopes and take a closer look at the coefficients appearing there.