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How to integrate a polynomial over a simplex. (English) Zbl 1216.68120

Summary: This paper starts by settling 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 explore our algorithms with some experiments. We conclude the article with extensions to other polytopes and discussion of other available methods.

MSC:

68Q25 Analysis of algorithms and problem complexity
52B55 Computational aspects related to convexity
68W30 Symbolic computation and algebraic computation
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