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Euler-Maclaurin with remainder for a simple integral polytope. (English) Zbl 1087.65002

A polytope in \(\mathbb R^n\) is called integral if its vertices are in the lattice \(\mathbb Z^n\); it is called simple if exactly \(n\) edges emanate from each vertex; it is called regular if the edges lie along lines that are generated by a \(\mathbb Z\)-basis of the lattice \(\mathbb Z^n\). The classical Euler-Maclaurin formula (EMF) computes the sum of the values of a function over the integer points in an interval. The first generalization concerns the sum of the values of polynomials or exponential functions on the convex integral and regular polytopes, next one generalizes it to simple integral polytopes.
Previously the authors proved an EMF with remainder for the sum of the values of an arbitrary smooth function on the lattice points in a regular polytopes. In the present article they generalize this formula to the case of simple integral lattice polytopes.

MSC:

65B15 Euler-Maclaurin formula in numerical analysis
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
52B55 Computational aspects related to convexity
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