On the Conjecture of Lehmer, limit Mahler measure of trinomials and asymptotic expansions

Jean-Louis Verger-Gaugry 1
1 Théorie des nombres
IF - Institut Fourier
Abstract : Let $n ≥ 2$ be an integer and denote by $\theta_n$ the real root in $(0, 1)$ of the trinomial $G_{n}(X) = −1 + X + X^n$ . The sequence of Perron numbers $(\theta_{n}^{−1} )_{n≥2}$ tends to 1. We prove that the Conjecture of Lehmer is true for $\{\theta_{n}^{−1} | n ≥ 2\}$ by the direct method of Poincar\'e asymptotic expansions (divergent formal series of functions) of the roots $\theta_n , z_{j,n}$, of $G_{n}(X)$ lying in $|z| < 1$, as a function of $n, j$ only. This method, not yet applied to Lehmer’s problem up to the knowledge of the author, is successfully introduced here. It first gives the asymptotic expansion of the Mahler measures ${\rm M}(G_n) = {\rm M}(\theta_{n}) = {\rm M}(\theta_{n}^{-1})$ of the trinomials $G_n$ as a function of $n$ only, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisot number. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. By this method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for $\{\theta_{n}^{−1} | n ≥ 2\}$, with a minoration of the house $\house\{\theta_{n}^{−1}\} = \theta_{n}^{−1}$ , and a minoration of the Mahler measure ${\rm M}(G_n)$ better than Dobrowolski’s one for $\{\theta_{n}^{−1} | n ≥ 2\}$ . The angular regularity of the roots of $G_n$ , near the unit circle, and limit equidistribution of the conjugates, for n tending to infinity (in the sense of Bilu, Petsche, Pritsker), towards the Haar measure on the unit circle, are described in the context of the Erd\H{o}s-Tur\'an-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions.
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Jean-Louis Verger-Gaugry. On the Conjecture of Lehmer, limit Mahler measure of trinomials and asymptotic expansions. 2015. ⟨hal-01196990⟩

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