Abstract. A period of rational integral is the result of integrating, with respect to
one or several variables, a rational function along a closed path. When
the period under consideration depends on a parameter, it satisfies a
specific linear differential equation called Picard-Fuchs equation. These
equations and their computation are important for computer algebra, but
also for algebraic geometry (they contains geometric invariants), in
combinatorics (many generating functions are periods) or in theoretical
physics. This thesis offers and studies algorithms to compute them.
The first chapter shows bounds on the size of Picard-Fuchs equations and on
the complexity of their computation. Existing algorithms for computing
these equations often produce, in the same time, certificates, typically
huge, which allows to check afterwards the correctness of the equation.
The bounds I obtained enlighten the computational nature of Picard-Fuchs
equations, they show in particular that the certificates are not a required
byproduct. The proof relies on the study of the generic case and the
reduction of pole order with Griffiths-Dwork method.
The second chapter offers an algorithms for computing Picard-Fuchs
equations more efficiently. It allows for the resolution of many
previously unsolved problems. It relies on a method for reducing the pole
order which extends Griffiths-Dwork reduction to the singulars cases.
The third chapter draws a rigorous correspondence between periods of
rational integrals and generating functions of multiple binomial sums.
Together with the computation of Picard-Fuchs equations, it allows
automatically proving identities about binomial sums.