Computing the Ehrhart quasi-polynomial of a rational simplex

Barvinok, Alexander

doi:10.1090/S0025-5718-06-01836-9

Computing the Ehrhart quasi-polynomial of a rational simplex
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Math. Comp. 75 (2006), 1449-1466 Request permission

Abstract:

We present a polynomial time algorithm to compute any fixed number of the highest coefficients of the Ehrhart quasi-polynomial of a rational simplex. Previously such algorithms were known for integer simplices and for rational polytopes of a fixed dimension. The algorithm is based on the formula relating the $k$th coefficient of the Ehrhart quasi-polynomial of a rational polytope to volumes of sections of the polytope by affine lattice subspaces parallel to $k$-dimensional faces of the polytope. We discuss possible extensions and open questions.

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