Asymptotic Euler-Maclaurin formula over lattice polytopes

TL;DR

This paper develops an asymptotic expansion formula for Riemann sums over lattice polytopes, extending the local Euler-Maclaurin formula and providing explicit terms and uniqueness results for Delzant polytopes.

Contribution

It introduces an independent proof of the asymptotic expansion formula for Delzant lattice polytopes and generalizes it to all lattice polytopes, with explicit formulas and uniqueness results.

Findings

Explicit formulas for expansion terms over Delzant polytopes in two dimensions

Third term formula for Delzant polytopes in arbitrary dimensions

Proof of uniqueness of the asymptotic expansion

Abstract

An asymptotic expansion formula of Riemann sums over lattice polytopes is given. The formula is an asymptotic form of the local Euler-Maclaurin formula due to Berline-Vergne. The proof given here for Delzant lattice polytopes is independent of the local Euler-Maclaurin formula. But we use it for general lattice polytopes. As corollaries, an explicit formula for each term in the expansion over Delzant polytopes in two dimension and an explicit formula for the third term of the expansion for Delzant polytopes in arbitrary dimension are given. Moreover, some uniqueness results are given.