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

International audienceLet $n ≥ 2$ 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 thatthe Conjecture of Lehmer is true for $\{\theta_{n}^{−1} | n ≥ 2\}$ by the direct method of Poincar\'e asymptoticexpansions (divergent formal series of functions) of the roots $\theta_n , z_{j,n}$, of $G_{n}(X)$ lying in $|z|