An algebraic construction of sum-integral interpolators

TL;DR

This paper develops an algebraic method to derive Euler-Maclaurin formulas for polytopes, unifying previous lattice point and Euler-Maclaurin formulas, with a combinatorial approach rooted in algebraic geometry.

Contribution

It introduces a new algebraic construction of sum-integral interpolators that generalizes existing formulas and is self-contained, avoiding reliance on toric geometry results.

Findings

Unified formulas for lattice points and Euler-Maclaurin sums

Algebraic construction of Todd classes and cycle intersections

Self-contained combinatorial approach

Abstract

This paper presents an algebraic construction of Euler-Maclaurin formulas for polytopes. The formulas obtained generalize and unite the previous lattice point formulas of Morelli and Pommersheim-Thomas, and the Euler-Maclaurin formulas of Berline-Vergne While the approach of this paper originates in the theory of toric varieties, and recovers previous results about characteristic classes of toric varieties, the present paper is self-contained and does not rely on results from toric geometry. We aim in particular to exhibit in a combinatorial way ingredients such as such Todd classes and cycle-level intersections in Chow rings, that first entered the theory of polytopes from algebraic geometry.