On the algebraic dependence of holonomic functions

Roques, Julien; Singer, Michael F.

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

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.

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.

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

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