On the algebraic dependence of holonomic functions

[Sur la dépendance algébrique des fonctions holonomes]

Roques, Julien¹ ; Singer, Michael F.²

¹ Université de Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, F-69622 Villeurbanne (France)
² Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC 27695-8205 (USA)

Annales Henri Lebesgue, Tome 5 (2022), pp. 141-177.

Abstract (VO)

We study the form of possible algebraic relations between functions satisfying linear differential equations. In particular, if $f$ and $g$ satisfy linear differential equations and are algebraically dependent, we give conditions on the differential Galois group associated to $f$ guaranteeing that $g$ is a polynomial in $f$. We apply this to hypergeometric functions and iterated integrals.

Résumé

Nous nous intéressons aux relations algébriques vérifiées par des solutions d’équations différentielles linéaires. En particulier, si $f$ et $g$ satisfont des équations différentielles linéaires et sont algébriquement dépendantes, nous donnons des conditions sur le groupe de Galois différentiel de $f$ garantissant que $g$ est un polynôme en $f$. Nous appliquons cela aux fonctions hypergéométriques et aux intégrales itérées.

Détail

Reçu le : 2021-01-26
Révisé le : 2021-07-24
Accepté le : 2021-09-04
Publié le : 2022-02-23

DOI : 10.5802/ahl.120

Classification : 12H05, 33C10, 34M03
Keywords: Linear Differential Equations, Differential Galois Theory, Algebraic Relations, Iterated Integrals, Hypergeometric Functions

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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