Vol. 318, No. 2, 2022

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An algebraic construction of sum-integral interpolators

Benjamin Fischer and Jamie Pommersheim

Vol. 318 (2022), No. 2, 305–338
Abstract

This paper presents an algebraic construction of the Euler–Maclaurin formulas for polytopes. The formulas obtained generalize and unite the previous lattice point formulas of Pommersheim and Thomas (2004) and Morelli (1993) and Euler–Maclaurin formulas of Berline and Vergne (2007) and Garoufalidis and Pommersheim (2012). While the approach of this paper originates in the theory of toric varieties and recovers results from Fischer and Pommersheim (2014) proved using toric geometry, 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 Todd classes and cycle-level intersections in Chow rings, that first entered the theory of polytopes from algebraic geometry.

Keywords
polytopes, Euler-Maclaurin summation, sum-integral interpolator, lattice point enumeration, toric variety, Todd class
Mathematical Subject Classification
Primary: 14M25, 52B20
Secondary: 11P21, 13F55
Milestones
Received: 12 January 2021
Revised: 25 April 2022
Accepted: 30 April 2022
Published: 20 August 2022
Authors
Benjamin Fischer
Department of Mathematics
Seattle Academy of Arts and Sciences
Seattle, WA
United States
Jamie Pommersheim
Mathematics Department
Reed College
Portland, OR
United States