Algebraic independence of $G$-functions and congruences “à la Lucas”

Adamczewski, Boris; Bell, Jason P.; Delaygue, Éric

doi:10.24033/asens.2392

Algebraic independence of

G

-functions and congruences “à la Lucas”
[Indépendance algébrique de

G

-fonctions et congruences « à la Lucas » ]

Adamczewski, Boris ; Bell, Jason P. ; Delaygue, Éric

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 3, pp. 515-559.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Résumé
Abstract

Nous développons une nouvelle méthode pour démontrer l'indépendance algébrique de $G$ -fonctions. Notre approche repose sur l'observation suivante : une $G$ -fonction est toujours solution d'une équation différentielle linéaire mais elle est aussi parfois solution d'une infinité d'équations aux différences linéaires associées au Frobenius que l'on obtient par réduction modulo des idéaux premiers. Lorsque ces équations aux différences linéaires sont d'ordre un, les coefficients de la $G$ -fonction correspondante satisfont des congruences rappelant un théorème classique de Lucas sur les coefficients binomiaux. Nous utilisons cette propriété pour en déduire un critère d'indépendance algébrique “à la Kolchin”. Nous montrons que ce critère est pertinent en démontrant que de nombreuses familles classiques de $G$ -fonctions satisfont des congruences “à la Lucas”.

We develop a new method for proving algebraic independence of $G$ -functions. Our approach rests on the following observation: $G$ -functions do not always come with a single linear differential equation, but also sometimes with an infinite family of linear difference equations associated with the Frobenius that are obtained by reduction modulo prime ideals. When these linear difference equations have order one, the coefficients of the corresponding $G$ -functions satisfy congruences reminiscent of a classical theorem of Lucas on binomial coefficients. We use this to derive a Kolchin-like criterion for algebraic independence. We show the relevance of this criterion by proving that many classical families of $G$ -functions turn out to satisfy congruences “à la Lucas”.

MR Zbl

DOI : 10.24033/asens.2392

Classification : 11J81; 11B85, 05.10, 12H99.
Keywords: Algebraic independence, $G$-functions, congruences, $p$-Lucas property, Kolchin's theorem, asymptotic
Mot clés : Indépendance algébrique, $G$-fonctions, congruences, propriété $p$-Lucas, théorème de Kolchin, asymptotique

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     author = {Adamczewski, Boris and Bell, Jason P. and Delaygue, \'Eric},
     title = {Algebraic independence of~$G$-functions and congruences {\textquotedblleft}\`a la {Lucas{\textquotedblright}}},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {515--559},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 52},
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     doi = {10.24033/asens.2392},
     mrnumber = {3982874},
     zbl = {1450.11075},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/asens.2392/}
}

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Adamczewski, Boris; Bell, Jason P.; Delaygue, Éric. Algebraic independence of $G$-functions and congruences “à la Lucas”. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 3, pp. 515-559. doi : 10.24033/asens.2392. http://archive.numdam.org/articles/10.24033/asens.2392/

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