Louis Dumont

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Thesis defense

TITLE: Efficient algorithms for the symbolic computation of some contour integrals depending on one parameter [pdf]

DATE: Monday, December 5th 2016 at 2:30 PM

PLACE: Alan Turing building, amphitheater Sophie Germain, on the campus of École polytechnique (how to get there)

WARNING: Because of the "Vigipirate" setup, visitors need to be equipped with some I.D. to enter the building.

In this thesis, we provide solutions to some symbolic integration problems in computer algebra. The main objective is to effectively and efficiently compute functions that appear as contour integrals depending on one parameter.

First, we consider the computation of the integral of a bivariate rational function with regard to one of the variables. The result is then an algebraic function that can be expressed as a sum of residues of the integrand. We design two algorithms that efficiently compute an annihilating polynomial for each residue, and then for their sum, which yields an annihilating polynomial for the integral itself.

These algorithms apply almost directly to the computation of an annihilating polynomial for the diagonal of a rational function, that is, the univariate power series obtained from the expansion of a bivariate rational function by only keeping the diagonal coefficients. Indeed, these diagonals can be written as integrals of rational functions. In another application, we give a new algorithm for the Taylor expansion of the generating functions for several families of unidimensional lattice walks. It relies on a fine analysis of the sizes of the algebraic and differential equations satisfied by these generating functions.

Secondly, we consider integrals of mixed hypergeometric and hyperexponential terms. In this case, the result is a polynomially recursive sequence. We devise a method to rewrite the various shifts of a given term under a normal form. This allows us to apply the method of reduction-based creative telescoping in order to efficiently compute a recurrence with polynomial coefficients for the integral.