Périodes d'intégrales rationnelles : algorithmes et applications

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.
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Submitted on : Monday, December 1, 2014 - 11:15:20 AM
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Pierre Lairez. Périodes d'intégrales rationnelles : algorithmes et applications. Calcul formel [cs.SC]. École polytechnique, 2014. Français. ⟨tel-01089130⟩

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