Computing periods of rational integrals

A period of a rational integral is the result of integrating, with respect to one or several variables, a rational function over a closed path. This work focuses particularly on periods depending on a parameter: in this case the period under consideration satisfies a linear differential equation, the Picard-Fuchs equation. I give a reduction algorithm that extends the Griffiths-Dwork reduction and apply it to the computation of Picard-Fuchs equations. The resulting algorithm is elementary and has been successfully applied to problems that were previously out of reach.Comment: To appear in Math. comp. Supplementary material at http://pierre.lairez.fr/supp/periods

Une approche par l’analyse algébrique effectivedes systèmes linéaires sur des algèbres de Ore

The purpose of this paper is to present a survey on the effective algebraic analysis approach to linear systems theory with applications to control theory and mathematical physics. In particular, we show how the combination of effective methods of computer algebra - based on Gröbner basis techniques over a class of noncommutative polynomial rings of functional operators called Ore algebras - and constructive aspects of module theory and homological algebra enables the characterization of structural properties of linear functional systems. Algorithms are given and a dedicated implementation, called OreAlgebraicAnalysis, based on the Mathematica package HolonomicFunctions, is demonstrated.Le but de ce papier est de présenter un état de l’art d’une approche par l’analyse algébrique effective de la théorie des systèmes linéaires avec des applications à la théorie du contrôle et à la physique mathématique.En particulier, nous montrons comment la combinaison des méthodes effectives de calcul formel - basées sur lestechniques de bases de Gröbner sur une classe d’algèbres polynomiales noncommutatives d’opérateurs fonctionnels appelée algèbres de Ore - et d’aspects constructifs de théorie des modules et d’algèbre homologique permet lacaractérisation de propriétés structurelles des systèmes linéaires fonctionnels. Des algorithmes sont donnés et uneimplémentation dédiée, appelée OREALGEBRAICANALYSIS, basée sur le package Mathematica HOLONOMIC-FUNCTIONS, est présenté

A generalization of Serre's conjecture and some related issues

AbstractSeveral topics concerned with multivariate polynomial matrices like unimodular matrix completion, matrix determinantal or primitive factorization, matrix greatest common factor existence and subsequent extraction along with relevant primeness and coprimeness issues are related to a conjecture which may be viewed as a type of generalization of the original Serre problem (conjecture) solved nonconstructively in 1976 and constructively, more recently. This generalized Serre conjecture is proved to be equivalent to several other unsettled conjectures and, therfore, all these conjectures constitute a complete set in the sense that solution to any one also solves all the remaining

Feynman integral relations from parametric annihilators

We study shift relations between Feynman integrals via the Mellin transform through parametric annihilation operators. These contain the momentum space IBP relations, which are well-known in the physics literature. Applying a result of Loeser and Sabbah, we conclude that the number of master integrals is computed by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate techniques to compute this Euler characteristic in various examples and compare it with numbers of master integrals obtained in previous works.Comment: v2: new section 3.1 added, several misprints corrected and additional remark

A Factorization Algorithm for G-Algebras and Applications

It has been recently discovered by Bell, Heinle and Levandovskyy that a large class of algebras, including the ubiquitous $G$-algebras, are finite factorization domains (FFD for short). Utilizing this result, we contribute an algorithm to find all distinct factorizations of a given element $f \in \mathcal{G}$, where $\mathcal{G}$ is any $G$-algebra, with minor assumptions on the underlying field. Moreover, the property of being an FFD, in combination with the factorization algorithm, enables us to propose an analogous description of the factorized Gr\"obner basis algorithm for $G$-algebras. This algorithm is useful for various applications, e.g. in analysis of solution spaces of systems of linear partial functional equations with polynomial coefficients, coming from $\mathcal{G}$. Additionally, it is possible to include inequality constraints for ideals in the input