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The Ehrhart polynomial of a lattice \(n\)-simplex. (English) Zbl 0871.52008

Summary: The problem of counting the number of lattice points inside a lattice polytope in \(\mathbb{R}^{n}\) has been studied from a variety of perspectives, including the recent work of Pommersheim and Kantor-Khovanskii using toric varieties and Cappell-Shaneson using Grothendieck-Riemann-Roch. Here we show that the Ehrhart polynomial of a lattice \(n\)-simplex has a simple analytical interpretation from the perspective of Fourier Analysis on the \(n\)-torus. We obtain closed forms in terms of cotangent expansions for the coefficients of the Ehrhart polynomial, that shed additional light on previous descriptions of the Ehrhart polynomial.

MSC:

52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
11P21 Lattice points in specified regions
11F20 Dedekind eta function, Dedekind sums
05A15 Exact enumeration problems, generating functions
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
11H06 Lattices and convex bodies (number-theoretic aspects)
14L24 Geometric invariant theory
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References:

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