On upper bounds for the number of minimal polynomials with bounded derivative at a root

D. V. Vasilyev ; A. S. Kudin

Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 47-54

Résumé

In the paper we consider the problem of obtaining estimates for the number of minimal integer polynomials $P(x)$ of degree $n$ and height not exceeding $Q$, such that the derivative is bounded at a root $\alpha$, i.e. $\left| P'(\alpha) \right| Q^{1-v}$ for some $v > 0$. This problem occurs naturally in many problems of metric number theory related to obtaining effective estimates for the measure of points at which integral polynomials from some class take small values. For example, in 1976 R. Baker has used such an estimate for obtaining an upper bound for the Hasdorff dimension in Baker-Schdimt problem. We prove that the number of polynomials $P(x)$ defined above having roots $\alpha$ on the interval $\left( -\frac12; \frac12 \right)$ doesn't exceed $c_1(n)Q^{n+1-\frac35 v}$ for $Q>Q_0(n)$ and $1.5 \le v \le \frac12 (n+1)$. The result is based on an imrovement to the lemma on small integer polynomial divisor extraction from A.O. Gelfond's monograph "Transcendetal and algebraic numbers".

Détail

Keywords: Diophantine approximation, Hausdorff dimension, transcendental numbers, resultant, Sylvester matrix, irreducible divisor, Gelfond's lemma.