Counting real algebraic numbers with bounded derivative of minimal polynomial

Abstract

In this paper, we consider the problem of counting algebraic numbers [Formula: see text] of fixed degree [Formula: see text] and bounded height [Formula: see text] such that the derivative of the minimal polynomial [Formula: see text] of [Formula: see text] is bounded, [Formula: see text]. This problem has many applications to the problems of metric theory of Diophantine approximation. We prove that the number of [Formula: see text] defined above on the interval [Formula: see text] does not exceed [Formula: see text] for [Formula: see text] and [Formula: see text]. Our result is based on an improvement to a lemma from Gelfond’s monograph “Transcendental and algebraic numbers”. Given an integer polynomial small enough in some point, the lemma provides an upper bound for the absolute value of its irreducible divisor. We obtain a stronger estimate which holds in real points located far enough from all algebraic numbers of bounded degree and height. This is done by considering the resultant of two polynomials represented as the determinant of the Sylvester matrix for the shifted counterparts.