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, Online first, 44 p.

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

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

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     title = {Limits of {Mahler} measures in multiple variables},
     journal = {Annales de l'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, Online first, 44 p.

Bibliographie
Cité par

[1] Akhtari, Shabnam; Vaaler, Jeffrey D. Lower bounds for Mahler measure that depend on the number of monomials, Int. J. Number Theory, Volume 15 (2019) no. 7, pp. 1425-1436 | DOI | MR | Zbl

[2] Apostol, Tom M. Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer, 1976, xii+338 pages | MR | Zbl

[3] Belabas, Karim; Cohen, Henri Numerical algorithms for number theory—using Pari/GP, Mathematical Surveys and Monographs, 254, American Mathematical Society, 2021, ix+429 pages | DOI | MR | Zbl

[4] Betke, U.; Henk, M.; Wills, J. M. Successive-minima-type inequalities, Discrete Comput. Geom., Volume 9 (1993) no. 2, pp. 165-175 | DOI | MR | Zbl

[5] Bleistein, Norman; Handelsman, Richard A. Asymptotic expansions of integrals, Dover Publications, 1986, xvi+425 pages | MR

[6] Boyd, David W. Kronecker’s theorem and Lehmer’s problem for polynomials in several variables, J. Number Theory, Volume 13 (1981) no. 1, pp. 116-121 | DOI | MR | Zbl

[7] Boyd, David W. Speculations concerning the range of Mahler’s measure, Can. Math. Bull., Volume 24 (1981) no. 4, pp. 453-469 | DOI | MR | Zbl

[8] Boyd, David W. Mahler’s measure and special values of $L$ -functions, Exp. Math., Volume 7 (1998) no. 1, pp. 37-82 | DOI | MR | Zbl

[9] Boyd, David W.; Mossinghoff, Michael J. Small limit points of Mahler’s measure, Exp. Math., Volume 14 (2005) no. 4, pp. 403-414 | DOI | MR | Zbl

[10] Boyd, David W.; Rodriguez-Villegas, Fernando Mahler’s measure and the dilogarithm. I, Can. J. Math., Volume 54 (2002) no. 3, pp. 468-492 | DOI | MR | Zbl

[11] Brunault, François; Zudilin, Wadim Many variations of Mahler measures. A lasting symphony, Australian Mathematical Society Lecture Series, 28, Cambridge University Press, 2020, xv+167 pages | DOI | MR | Zbl

[12] Cai, Xuanting; Todd, Robert G. A cellular basis for the generalized Temperley-Lieb algebra and Mahler measure, Topology Appl., Volume 178 (2014), pp. 107-124 | DOI | MR | Zbl

[13] Cartan, Henri Théorie élémentaire des fonctions analytiques d’une ou plusieurs variables complexes, Enseignement des Sciences, Hermann, 1961, 232 pages (avec le concours de Reiji Takahashi) | MR | Zbl

[14] Carter, Annie; Lalín, Matilde; Manes, Michelle; Miller, Alison Beth; Mocz, Lucia Two-variable polynomials with dynamical Mahler measure zero, Res. Number Theory, Volume 8 (2022) no. 2, 25, 22 pages | DOI | MR | Zbl

[15] Champanerkar, Abhijit; Kofman, Ilya On the Mahler measure of Jones polynomials under twisting, Algebr. Geom. Topol., Volume 5 (2005), pp. 1-22 | DOI | MR | Zbl

[16] Champanerkar, Abhijit; Kofman, Ilya On links with cyclotomic Jones polynomials, Algebr. Geom. Topol., Volume 6 (2006), pp. 1655-1668 | DOI | MR | Zbl

[17] Condon, John D. Asymptotic expansion of the difference of two Mahler measures, J. Number Theory, Volume 132 (2012) no. 9, pp. 1962-1983 | DOI | MR | Zbl

[18] D’Andrea, Carlos; Lalín, Matilde On the Mahler measure of resultants in small dimensions, J. Pure Appl. Algebra, Volume 209 (2007) no. 2, pp. 393-410 | DOI | MR | Zbl

[19] Deninger, Christopher Mahler measures and Fuglede-Kadison determinants, Münster J. Math., Volume 2 (2009), pp. 45-63 | MR | Zbl

[20] Dimitrov, Vesselin; Habegger, Philipp Galois orbits of torsion points near atoral sets (2019) (https://arxiv.org/abs/1909.06051) | DOI

[21] Dobrowolski, Edward On a question of Schinzel about the length and Mahler’s measure of polynomials that have a zero on the unit circle, Acta Arith., Volume 155 (2012) no. 4, pp. 453-463 | DOI | MR | Zbl

[22] Dobrowolski, Edward A note on Lawton’s theorem, Can. Math. Bull., Volume 60 (2017) no. 3, pp. 484-489 | DOI | MR | Zbl

[23] Dobrowolski, Edward; Smyth, Chris J. Mahler measures of polynomials that are sums of a bounded number of monomials, Int. J. Number Theory, Volume 13 (2017) no. 6, pp. 1603-1610 | DOI | MR | Zbl

[24] Dubickas, Artūras On sums of two and three roots of unity, J. Number Theory, Volume 192 (2018), pp. 65-79 | DOI | MR | Zbl

[25] Dubickas, Artūras; Jankauskas, Jonas Nonreciprocal algebraic numbers of small Mahler’s measure, Acta Arith., Volume 157 (2013) no. 4, pp. 357-364 | DOI | MR | Zbl

[26] Duke, W. A combinatorial problem related to Mahler’s measure, Bull. Lond. Math. Soc., Volume 39 (2007) no. 5, pp. 741-748 | DOI | MR | Zbl

[27] El Otmani, S.; Rhin, G.; Sac-Épée, J.-M. Finding new limit points of Mahler’s measure by genetic algorithms, Exp. Math., Volume 28 (2019) no. 2, pp. 129-131 | DOI | MR | Zbl

[28] Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. Higher transcendental functions. Vol. III, Robert E. Krieger Publishing Co., 1981, xvii+292 pages (based on notes left by Harry Bateman, reprint of the 1955 original) | MR | Zbl

[29] Everest, Graham; Ward, Thomas Heights of polynomials and entropy in algebraic dynamics, Universitext, Springer, 1999, xii+211 pages | DOI | MR | Zbl

[30] Gon, Yasuro; Oyanagi, Hideo Generalized Mahler measures and multiple sine functions, Int. J. Math., Volume 15 (2004) no. 5, pp. 425-442 | DOI | MR | Zbl

[31] Gu, Jarry; Lalín, Matilde The Mahler measure of a three-variable family and an application to the Boyd-Lawton formula, Res. Number Theory, Volume 7 (2021) no. 1, 13, 23 pages | DOI | MR | Zbl

[32] Guilloux, Antonin Experimentation around speed of convergence in Boyd-Lawton theorem for Mahler measure (2021) (https://gitlab.inria.fr/aguillou/mahler_condon)

[33] Guilloux, Antonin; Marché, Julien Volume function and Mahler measure of exact polynomials, Compos. Math., Volume 157 (2021) no. 4, pp. 809-834 | DOI | MR | Zbl

[34] Habegger, Philipp The norm of Gaussian periods, Q. J. Math., Volume 69 (2018) no. 1, pp. 153-182 | DOI | MR | Zbl

[35] Hajli, Mounir A new formula for Mahler’s measure, Funct. Approximatio, Comment. Math., Volume 62 (2020) no. 2, pp. 165-170 | DOI | MR | Zbl

[36] Issa, Zahraa; Lalín, Matilde A generalization of a theorem of Boyd and Lawton, Can. Math. Bull., Volume 56 (2013) no. 4, pp. 759-768 | DOI | MR | Zbl

[37] Kurokawa, N.; Lalín, Matilde; Ochiai, H. Higher Mahler measures and zeta functions, Acta Arith., Volume 135 (2008) no. 3, pp. 269-297 | DOI | MR | Zbl

[38] Lalín, Matilde; Sinha, Kaneenika Higher Mahler measure for cyclotomic polynomials and Lehmer’s question, Ramanujan J., Volume 26 (2011) no. 2, pp. 257-294 | DOI | MR | Zbl

[39] Lawton, Wayne M. A problem of Boyd concerning geometric means of polynomials, J. Number Theory, Volume 16 (1983) no. 3, pp. 356-362 | DOI | MR | Zbl

[40] Le, Thang Homology torsion growth and Mahler measure, Comment. Math. Helv., Volume 89 (2014) no. 3, pp. 719-757 | DOI | MR | Zbl

[41] Lehmer, D. H. Factorization of certain cyclotomic functions, Ann. Math., Volume 34 (1933) no. 3, pp. 461-479 | DOI | MR | Zbl

[42] Lind, Douglas; Schmidt, Klaus; Ward, Tom Mahler measure and entropy for commuting automorphisms of compact groups, Invent. Math., Volume 101 (1990) no. 3, pp. 593-629 | DOI | MR | Zbl

[43] Liu, Hang; Qin, Hourong Mahler measure of families of polynomials defining genus 2 and 3 curves, Exp. Math., Volume 32 (2023) no. 2, pp. 321-336 | DOI | MR | Zbl

[44] Lück, Wolfgang Estimates for spectral density functions of matrices over $ℂ [ℤ^{d}]$ , Ann. Math. Blaise Pascal, Volume 22 (2015) no. 1, pp. 73-88 | DOI | MR | Zbl

[45] Lück, Wolfgang Twisting $L^{2}$ -invariants with finite-dimensional representations, J. Topol. Anal., Volume 10 (2018) no. 4, pp. 723-816 | DOI | MR | Zbl

[46] Mahler, K. On some inequalities for polynomials in several variables, J. Lond. Math. Soc., Volume 37 (1962), pp. 341-344 | DOI | MR | Zbl

[47] Mehrabdollahei, Mahya Mahler measure of $P_{d}$ polynomials (2021) (https://arxiv.org/abs/2101.07675)

[48] Mellit, Anton Elliptic dilogarithms and parallel lines, J. Number Theory, Volume 204 (2019), pp. 1-24 | DOI | MR | Zbl

[49] Navot, Israel A Further Extension of the Euler-Maclaurin Summation Formula, Journal of Mathematics and Physics, Volume 41 (1962), pp. 155-163 | DOI | Zbl

[50] Pierce, Tracy A. The numerical factors of the arithmetic forms $\prod_{i = 1}^{n} (1 \pm α_{i}^{m})$ , Ann. Math., Volume 18 (1916) no. 2, pp. 53-64 | DOI | MR

[51] Raimbault, Jean Exponential growth of torsion in abelian coverings, Algebr. Geom. Topol., Volume 12 (2012) no. 3, pp. 1331-1372 | DOI | MR | Zbl

[52] Rudin, Walter Real and complex analysis, McGraw-Hill, 1987, xiv+416 pages | MR | Zbl

[53] Schinzel, A. On the Mahler measure of polynomials in many variables, Acta Arith., Volume 79 (1997) no. 1, pp. 77-81 | DOI | MR | Zbl

[54] Silver, Daniel S.; Williams, Susan G. Mahler measure of Alexander polynomials, J. Lond. Math. Soc., Volume 69 (2004) no. 3, pp. 767-782 | DOI | MR | Zbl

[55] Silver, Daniel S.; Williams, Susan G. Twisted Alexander invariants of twisted links, J. Knot Theory Ramifications, Volume 21 (2012) no. 11, 1250118, 17 pages | DOI | MR | Zbl

[56] Smyth, Chris J. An explicit formula for the Mahler measure of a family of 3-variable polynomials, J. Théor. Nombres Bordeaux, Volume 14 (2002) no. 2, pp. 683-700 | DOI | Numdam | MR | Zbl

[57] Smyth, Chris J. The Mahler measure of algebraic numbers: a survey, Number theory and polynomials (London Mathematical Society Lecture Note Series), Volume 352, Cambridge University Press, 2008, pp. 322-349 | DOI | MR | Zbl

[58] Smyth, Chris J. Closed sets of Mahler measures, Proc. Am. Math. Soc., Volume 146 (2018) no. 6, pp. 2359-2372 | DOI | MR | Zbl

[59] Vicente Gonçalves, J. L’inégalité de W. Specht, Univ. Lisboa Rev. Fac. Ci. A (2), Volume 1 (1950), pp. 167-171 | MR | Zbl

[60] Wiener, Norbert Tauberian theorems, Ann. Math., Volume 33 (1932) no. 1, pp. 1-100 | DOI | MR | Zbl

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