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arXiv · 0809.2083

How to Integrate a Polynomial over a Simplex

Velleda Baldoni·Nicole Berline·Jesus De Loera·Matthias Köppe·Michèle Vergne
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Abstract

This paper settles the computational complexity of the problem of integrating a polynomial function f over a rational simplex. We prove that the problem is NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin and Straus. On the other hand, if the polynomial depends only on a fixed number of variables, while its degree and the dimension of the simplex are allowed to vary, we prove that integration can be done in polynomial time. As a consequence, for polynomials of fixed total degree, there is a polynomial time algorithm as well. We conclude the article with extensions to other polytopes, discussion of other available methods and experimental results.

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Velleda Baldoni, Nicole Berline, Jesus De Loera, Matthias Köppe, Michèle Vergne. 2008-09-11. How to Integrate a Polynomial over a Simplex. https://doi.org/10.1090/s0025-5718-2010-02378-6

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