Limits of Mahler measures in multiple variables

Brunault, François; Guilloux, Antonin; Mehrabdollahei, Mahya; Pengo, Riccardo

doi:10.5802/aif.3611

Limits of Mahler measures in multiple variables
[Limites de mesures de Mahler en plusieurs variables]

Brunault, François ¹ ; Guilloux, Antonin ² ; Mehrabdollahei, Mahya ² ; Pengo, Riccardo ³

¹ UMPA, École normale supérieure de Lyon, 46 allée d’Italie, 69100 Lyon (France)
² IMJ-PRG, Sorbonne Université, CNRS and OURAGAN, INRIA, 4 place Jussieu, Boite Courrier 247, 75252 Paris Cedex 5 (France)
³ Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn (Germany)

Annales de l'Institut Fourier, Tome 74 (2024) no. 4, pp. 1407-1450.

Résumé
Abstract

Nous prouvons que certaines suites de polynômes de Laurent, obtenues à partir d’un polynôme de Laurent $P$ fixé par substitutions monomiales, donnent des suites de mesures de Mahler qui convergent vers la mesure de Mahler de $P$ . Ce résultat généralisent des travaux antérieurs de Boyd et Lawton, qui considéraient des substitutions univariées. Nous obtenons aussi une borne supérieure explicite pour le terme d’erreur dans cette convergence, généralisant des travaux de Dimitrov et Habegger. Nous donnons enfin un développement asymptotique complet pour une famille particulière de polynômes bivariés, dont les mesures de Mahler avaient été étudiées indépendamment par la troisième autrice.

We prove that certain sequences of Laurent polynomials, obtained from a fixed multivariate Laurent polynomial $P$ by monomial substitutions, give rise to sequences of Mahler measures which converge to the Mahler measure of $P$ . This generalises previous work of Boyd and Lawton, who considered univariate monomial substitutions. We provide moreover an explicit upper bound for the error term in this convergence, extending work of Dimitrov and Habegger, and a full asymptotic expansion for a family of $2$ -variable polynomials, whose Mahler measures were studied independently by the third author.

Reçu le : 2022-06-27
Accepté le : 2022-11-17
Première publication : 2024-05-17
Publié le : 2024-07-03

DOI : 10.5802/aif.3611

Classification : 11R06
Keywords: Mahler Measure, Monomial substitutions.
Mot clés : Measure de Mahler, substitutions monomiales.

Affiliations des auteurs :

Brunault, François ¹ ; Guilloux, Antonin ² ; Mehrabdollahei, Mahya ² ; Pengo, Riccardo ³

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Limits of {Mahler} measures in multiple variables},
     journal = {Annales de l'Institut Fourier},
     pages = {1407--1450},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Brunault, François; Guilloux, Antonin; Mehrabdollahei, Mahya; Pengo, Riccardo. Limits of Mahler measures in multiple variables. Annales de l'Institut Fourier, Tome 74 (2024) no. 4, pp. 1407-1450. doi : 10.5802/aif.3611. https://aif.centre-mersenne.org/articles/10.5802/aif.3611/

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