On real algebraic numbers in which the derivative of their minimal polynomial is small

D V Koleda

doi: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]

Highlights

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

Summary

Introduction

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. Теорема 1 внешне напоминает результат из [7], где при n ≥ 2 показано, что для любого промежутка I ⊆ количество Φ n (Q, I ) алгебраических чисел α ∈ I степени n и высоты ≤Q удовлетворяет соотношению В упомянутой выше работе Бэйкера [1] для целого n ≥ 1 и вещественных H ≥ 1 рассматривается множество n*(H , X ) неприводимых над многочленов P степени deg P = n и высоты H(P) = H, которые имеют взаимно простые коэффициенты и старший коэффициент an = H, и у которых есть корень α ∈ с условием | P′(α) |< X , а также подразумеваются некоторые технические ограничения, связанные с решаемой в [1] задачей.

Results

Conclusion