Sum-integral interpolators and the Euler-Maclaurin formula for polytopes
type: Item
instance of: scholarly article
title: Sum-integral interpolators and the Euler-Maclaurin formula for polytopes
zbMATH Open document ID: 1266.14040
DOI: 10.1090/S0002-9947-2012-05381-5
author: Stavros Garoufalidis
author: James Pommersheim
published in: Transactions of the American Mathematical Society
publication date: 2012-05-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1002.3522
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 6038804
zbMATH Keywords: lattice polytopes
zbMATH Keywords: Euler-MacLaurin formula
zbMATH Keywords: flag varieties
zbMATH Keywords: exponential sums
zbMATH Keywords: interpolators
MaRDI profile type: Publication
arXiv ID: 1002.3522
cites work: A Polynomial Time Algorithm for Counting Integral Points in Polyhedra When the Dimension is Fixed
cites work: Q4938470
cites work: Q5447543
cites work: Points entiers dans les polyèdres convexes
cites work: Introduction to Toric Varieties. (AM-131)
cites work: Q4011317
cites work: Lattice invariant valuations on rational polytopes
cites work: Pick's theorem and the Todd class of a toric variety
cites work: Cycles representing the Todd class of a toric variety
cites work: Cycle-Level Intersection Theory for Toric Varieties
OpenAlex ID: W2962790829
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review text: If \(P\) is a lattice polyhedron in a rational vector space \(V\) and \(f\) is a polynomial function defined on \(P\), then one is interested in nice expressions for the sum \(s_f(P)\) of all \(f(x)\) where \(x\) runs through all lattice points of \(P\). In particular, such a formula is called ``local Euler-Maclaurin'', if it expresses \(s_f(P)\) as a sum over all faces \(F\) of \(P\) of integrals of the form \(\int_F D(P,F)\cdot f\). Here \(D(P,F)\) is a differential operator of infinite order, but with constant coefficients that acts as a kind of a weight working on \(f\). The striking point is the locality, i.e.\ \(D(P,F)\) is supposed to depend only on the (conical) shape of \(P\) in a neighborhood of a general point of \(F\).NEWLINENEWLINEThose formulae are known to exist (McMullen), but the operators are not uniquely determined. The choice can become canonical if one uses some additional structure like a scalar product on \(V\) or the presence of a complete flag in \(V^*\). In the present paper, the authors obtain a solution of a similar problem involving exponential sums arising as the kernels of the Laplace transformation. Here, the weight becomes a true function \(\mu(P,F)\) called interpolator, and knowing \(\mu\) does even imply solutions of the above Euler-Maclaurin problem and its inverse.NEWLINENEWLINEMoreover, the interpolator \(\mu\) is effectively computable, and it becomes canonical if understood as being parametrized by the flag variety on \(V^*\).
reviewed by: Klaus Altmann
description: scientific article; zbMATH DE number 6038804