In this paper we consider the problem of counting algebraic numbers α of fixed degree n and bounded height Q such that the derivative of the minimal polynomial Pα(x) of α is bounded,This problem has many applications to the problems of the metric theory of Diophantine approximation. We prove that the number of α defined above on the interval − 1 2 , 1 2 doesn't exceed c1(n)Q n+1− 1 7 v for Q > Q0(n) and 1.4 ≤ v ≤ 7 16 (n + 1). Our result is based on an improvement to the lemma on the order of zero approximation by irreducible divisors of integer polynomials from A. Gelfond's monograph "Transcendental and algebraic numbers". The improvement provides a stronger estimate for the absolute value of the divisor in real points which are located far enough from all algebraic numbers of bounded degree and height and it's based on the representation of the resultant of two polynomials as the determinant of Sylvester matrix for the shifted polynomials.

In the metric theory of Diophantine approximations, one of the main problems leading to exact characteristics in the classifications of Mahler and Koksma is to estimate the Lebesgue measure of the points x ∈ B ⊂ I from the interval I such as the inequality | P (x) | < Q-w, w > n, Q >1 for the polynomials P(x) ∈ Z[x], deg P ≤ n, H(P) ≤Q is satisfied. The methods of obtaining estimates are different at different intervals of w change. In this article, at w > n +1 we get the estimate µ B< c1(n) Q – (w-1/n). The best estimate to date was c2(n) Q –(w- n/n).