Sum-integral interpolators and the Euler-Maclaurin formula for polytopes

TL;DR

This paper develops a new framework for local lattice point counting and Euler-Maclaurin formulas for polytopes using interpolators derived from rigid complement maps, unifying previous approaches.

Contribution

It introduces the concept of interpolators between exponential sums and integrals, providing a unified, effective method to derive local Euler-Maclaurin formulas for rational polytopes.

Findings

Constructs effectively computable interpolators from rigid complement maps.

Generalizes and unifies previous lattice point counting formulas.

Provides new local Euler-Maclaurin formulas for polytopes.

Abstract

A local lattice point counting formula, and more generally a local Euler-Maclaurin formula follow by comparing two natural families of meromorphic functions on the dual of a rational vector space $V$, namely the family of exponential sums (S) and the family of exponential integrals (I) parametrized by the set of rational polytopes in $V$. The paper introduces the notion of an interpolator between these two families of meromorphic functions. We prove that every rigid complement map in $V$ gives rise to an effectively computable \SI-interpolator (and a local Euler-MacLaurin formula), an \IS-interpolator (and a reverse local Euler-MacLaurin formula) and an \ISo-interpolator. Rigid complement maps can be constructed by choosing an inner product on $V$ or by choosing a complete flag in $V$. The corresponding interpolators generalize and unify the work of Berline-Vergne, Pommersheim-Thomas, and Morelli.