Mahler measure of P d polynomials
Mahya Mehrabdollahei1
1 IMJ-PRG, Sorbonne Université 4 Pl. Jussieu, Paris, France
Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 795-817.

Nous étudions la mesure de Mahler d’une famille de polynômes à deux variables, désignée par P d avec d≥1, non bornée en degré et en genre. En utilisant une formule de forme close de la mesure de Mahler [13], nous sommes capables de calculer m(P d ), avec d arbitraire, comme une somme des valeurs du dilogarithme en certaines racines de l’unité. Nous prouvons que m(P d ) converge et que la limite est un multiple de ζ(3), où ζ est la fonction zêta de Riemann. La preuve que nous donnons est computationnelle et est basée sur l’estimation de l’erreur des sommes de Riemann d’une fonction bivariée. Nous exposons une deuxième preuve possible et plus courte basée sur une généralisation conjecturée du théorème de Boyd–Lawton et sur un résultat de D’Andrea et Lalín [11].

This article investigates the Mahler measure of a family of 2-variate polynomials, denoted by P d , for d≥1, unbounded in both degree and genus. By using a closed formula for the Mahler measure [13], we are able to compute m(P d ), for arbitrary d, as a sum of the values of dilogarithm at special roots of unity. We prove that m(P d ) converges, and the limit is proportional to ζ(3), where ζ is the Riemann zeta function. The proof we give is computational and based on the estimation of the error of Riemann sums of a bivariate function. We describe a second possible shorter proof based on a conjectural generalization of the theorem of Boyd–Lawton and a result of D’Andrea and Lalín [11].

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DOI : 10.5802/jtnb.1264
Classification : 11R06
Mots clés : Mahler measure, polynomial, exact polynomial, Bloch–Wigner dilogarithm, Riemann sums
Mahya Mehrabdollahei 1

1 IMJ-PRG, Sorbonne Université 4 Pl. Jussieu, Paris, France
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Mahya Mehrabdollahei. Mahler measure of $P_d$ polynomials. Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 795-817. doi : 10.5802/jtnb.1264. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1264/

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