A Dobrowolski type minoration of the Mahler measure of height 1 trinomials
Résumé
The method of Poincaré asymptotic expansions on the roots of the trinomials G_n (x) := -1+x+x^n is used to deduce the limit Mahler mesure lim_{n \to \infty} M(G_n) = 1.38135 =: Lambda, as a Log-Sine integral (Clausen integral, Bloch-Wigner dilogarithm). This method is compared with that of Boyd and Smyth with 2-variables Mahler measures, who have also proved that this limit is Lambda by their approach. The present method allows to obtain an asymptotic expansion of M(G_n) with respect to Lambda, and a Dobrowolski-type inequality for $\theta_n^{-1}$, as a function of n, if $\theta_n$ is the root of G_n in the interval (0,1). The extension to a Conjecture of C. Smyth on the limit Mahler measure of the trinomials
\pm 1 \pm x^k + x^n , as n tends to infinity, is discussed, by comparison with the recent result of Flammang stating that this Conjecture is true for large n.
Domaines
Théorie des nombres [math.NT]
Origine : Fichiers produits par l'(les) auteur(s)