Limits of Mahler measures in multiple variables
Annales de l'Institut Fourier, Online first, 44 p.

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.

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.

Received:
Accepted:
Online First:
DOI: 10.5802/aif.3611
Classification: 11R06
Keywords: Mahler Measure, Monomial substitutions.
Mot clés : Measure de Mahler, substitutions monomiales.
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)
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Brunault, François; Guilloux, Antonin; Mehrabdollahei, Mahya; Pengo, Riccardo. Limits of Mahler measures in multiple variables. Annales de l'Institut Fourier, Online first, 44 p.

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