Residues formulae for volumes and Ehrhart polynomials of convex polytopes

TL;DR

This paper introduces residue formulae for calculating volumes and Ehrhart polynomials of convex polytopes, applying these methods to specific polytopes like the Chan-Robbins and Pitman-Stanley polytopes.

Contribution

It develops residue-based formulas for volumes and Ehrhart polynomials, providing new computational tools for convex polytopes and applying them to notable examples.

Findings

Residue formulae for convex polytope volumes

Residue-based Ehrhart polynomial formulas

Application to Chan-Robbins and Pitman-Stanley polytopes

Abstract

In these notes, we explain residue formulae for volumes of convex polytopes, and for Ehrahrt polynomials based on the notion of total residue. We apply this method to the computation of the volume of the Chan-Robbins polytope. The final computation is based on a total residue formula for the system $A_n$, similar to Morris identity. For flow polytopes, a formula of change of variables in total residues leads to a "nice formula" for Ehrhart polynomials in function of mixed volumes. We apply it to Pitman-Stanley polytope.