On real algebraic numbers in which the derivative of their minimal polynomial is small
https://doi.org/10.29235/1561-2430-2021-57-2-135-147
Abstract
Algebraic numbers are the roots of integer polynomials. Each algebraic number α is characterized by its minimal polynomial Pα that is a polynomial of minimal positive degree with integer coprime coefficients, α being its root. The degree of α is the degree of this polynomial, and the height of α is the maximum of the absolute values of the coefficients of this polynomial. In this paper we consider the distribution of algebraic numbers α whose degree is fixed and height bounded by a growing parameter Q, and the minimal polynomial Pα is such that the absolute value of its derivative P'α (α) is bounded by a given parameter X. We show that if this bounding parameter X is from a certain range, then as Q → +∞ these algebraic numbers are distributed uniformly in the segment [-1+√2/3.1-√2/3]
About the Author
D. V. Koleda
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Denis V. Koleda – Ph. D. (Physics and Mathematics), Senior Researcher of the Department of Number Theory
11, Surganov Str., 220072, Minsk, Republic of Belarus
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