# Louis Dumont

Home
| Research | Publications
| Thesis defense |
My
PGP key

## 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.

ABSTRACT :

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.