Residue formulae, vector partition functions and lattice points in rational polytopes. (English) Zbl 0926.52016

Brion, Michel; Vergne, Michèle

J. Am. Math. Soc. 10, No. 4, 797-833 (1997).

The authors develop residue formulae for a class of functions of several variables and apply these to various problems. In particular they obtain closed formulae for vector partition functions and for their continuous analogs. These imply an Euler-MacLaurin summation formula for vector partition functions and also for rational convex polytopes.
So the authors express the sum of values of a polynomial function at all lattice points of a rational convex polytope in terms of the variation of the integral of the function over the deformed polytope.

Reviewer: Jörg M.Wills (Siegen)

MSC:

52B20	Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
11P21	Lattice points in specified regions
52C07	Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)

Keywords:

lattice points; lattice polytopes; vector partition functions; rational convex polytopes

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