How to Integrate a Polynomial over a Simplex

Details How to Integrate a Polynomial over a Simplex How to Integrate a Polynomial over a Simplex

2008-09-11

arxiv.org/abs/0809.2083v3

Mathematics

Metric Geometry

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Scientific paper

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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