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Mathematics of Computation

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How to integrate a polynomial over a simplex


Authors: Velleda Baldoni, Nicole Berline, Jesus A. De Loera, Matthias Köppe and Michèle Vergne
Journal: Math. Comp. 80 (2011), 297-325
MSC (2010): Primary 68W30; Secondary 52B55
DOI: https://doi.org/10.1090/S0025-5718-2010-02378-6
Published electronically: July 14, 2010
Table supplement: Appendix A
MathSciNet review: 2728981
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Abstract: 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 $ \mathrm{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.


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Additional Information

Velleda Baldoni
Affiliation: Dipartimento di Matematica, Università degli studi di Roma “Tor Vergata”, Via della ricerca scientifica 1, I-00133, Italy
Email: baldoni@mat.uniroma2.it

Nicole Berline
Affiliation: Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau Cedex, France
Email: nicole.berline@math.polytechnique.fr

Jesus A. De Loera
Affiliation: Department of Mathematics, University of California, Davis, One Shields Avenue, Davis, California 95616
Email: deloera@math.ucdavis.edu

Matthias Köppe
Affiliation: Department of Mathematics, University of California, Davis, One Shields Avenue, Davis, California 95616
Email: mkoeppe@math.ucdavis.edu

Michèle Vergne
Affiliation: Institut de Mathématiques de Jussieu, 175 rue du Chevaleret, 75103 Paris, France
Email: vergne@math.jussieu.fr

DOI: https://doi.org/10.1090/S0025-5718-2010-02378-6
Received by editor(s): February 16, 2009
Received by editor(s) in revised form: September 25, 2009
Published electronically: July 14, 2010
Article copyright: © Copyright 2010 American Mathematical Society