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
