How to integrate a polynomial over a simplex

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 poly-nomials 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 and discussion of other available methods.