Section: New Results

Integration of rational functions

Periods of rational integrals are specific integrals, with respect to one or several variables, whose integrand is a rational function and whose domain of integration is closed. This particular class of integrals contains large families of functions naturally occurring in combinatorics and statistical physics, such as diagonals, constant terms and positive part of rational functions. Periods involving one parameter are classically known to satisfy Picard-Fuchs equations, a special type of linear differential equations with a very rich analytic and arithmetic structure. As for other special-function manipulations, handling periods through those differential equations is a good way to actually compute them, and this was the topic of Pierre Lairez' PhD thesis defended in 2014  [53] and awarded the “Ecole Polytechnique thesis prize” in 2015.

Computing multivariate integrals is one speciality of the team and our algorithms are known to treat much more general integrals than just periods of rational integrals. However, integration is still slow in practice when the number of variables goes increasing. By looking at periods of rational functions, the hope is to obtain relevant complexity bounds and faster algorithms.

The goal of reaching relevant theoretical complexity bounds had been reached in 2013  [31] but a practically fast algorithm was still missing. This year, we described a new algorithm which is efficient in practice [4] , though its complexity is not known. This algorithm allows to compute quickly integrals that are too big to be computed with previous algorithms. As a challenging benchmark, we computed 210 integrals given by Batyrev and Kreuzer in their work on Calabi–Yau varieties. This achievement gave strong visibility to the paper and allowed a quick dissemination of the implementation, which is provided in Magma under a CeCILL B license. The algorithm is now used on a regular basis by several teams. We know of:

  • Tom Coates' team (Dpt. of Mathematics, Imperial College, London, UK), which uses the software in their work about mirror symmetry and classification of Fano varieties;

  • Duco van Straten (Institute of Mathematics, University of Mainz, Germany), who uses the software in his work in algebraic geometry;

  • Gert Alkmvist (Dpt. of Mathematics, University of Lund, Sweden), who uses the software in his work of enumerating the Calabi–Yau differential equations.