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.

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