A Panorama on the Minoration of the Mahler Measure: from the Problem of Lehmer to its Reformulations in Topology and Geometry

Jean-Louis Verger-Gaugry (1)

1 LAMA - Laboratoire de Mathématiques

This text is addressed to readers who are already engaged in their own research in the problem of Lehmer in at least one domain concerned, like graduate students, post-docs and active researchers. Initially the problem of Lehmer appeared in the original paper of D. Lehmer in Number Theory in 1933. The great richness of the problem of Lehmer is reflected today by the fact that the problem of minoration of the Mahler measure can been reformulated in several domains. Many interesting directions of research have appeared in doing so, and are developping in parallel. These domains are evoked in the present article. The reader not accustomed to the problem of Lehmer may find an interest to discover the different reformulations of the Mahler measure and its realizations in topology, geometry,... The aims of this text is to provide an extremely sketchy panorama of a large area of this part of mathematics, now spread over Number Theory, Arithmetic Geometry, and much more. It can be used as a guide to those who want to embark on a more serious study of one of these domains, or develop new analogues in existing theories. The Conjecture of Lehmer, and its refinement in 1965 by Schinzel, the Conjecture ofSchinzel-Zassenhaus, amounts to a problem of universal minoration of the Mahler mea-sure, and of the height in higher dimension in Arithmetic Geometry. The objective ofthis Survey is to review the numerous minorations obtained in these two domains, in par-ticular Dobrowolski’s inequality, then to present the analogues of the problem of Lehmerin different contexts with various analogues of the Mahler measure and the height.The reformulation of the problem of Lehmer in other domains brings to light a certainnumber of situations generating integer polynomials for which the Problem of Lehmer isasked, and, if a nontrivial lower bound exists to the Mahler measure of these polynomials,the meaning and the realization of the situation of extremality. In several cases Lehmer’snumber is found to be a nontrivial minorant and is shown to be reached.

Mots clés: Lehmer conjecture; Lehmer problem; Lehmer's number; Schinzel-Zassenhaus conjecture; mapping class; stretch factor; coxeter system; Salem number; Perron number; Pisot number; Mahler measure; minoration; algebraic dynamical system; torsion growth; heights; knots; links; Alexander polynomial; Jones polynomial

Domaines: Théorie des nombres [math.NT]

Jean-Louis Verger-Gaugry. A Panorama on the Minoration of the Mahler Measure: from the Problem of Lehmer to its Reformulations in Topology and Geometry. 2020. ⟨hal-03148129⟩