danДоклады Национальной академии наук БеларусиDoklady of the National Academy of Sciences of Belarus1561-83232524-2431The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-8323-2026-70-4-271-279dan-1318Research ArticleМАТЕМАТИКАMATHEMATICSО распределении алгебраических чисел, имеющих заданную степень над числовым полемOn the distribution of algebraic numbers having a given degree over a number fieldКоледаД. В.KoledaD. V.

Коледа Денис Владимирович – канд. физ.-мат. наук, ст. науч. сотрудник

ул. Сурганова, 11, 220072, Минск 

Koleda Denis V. – Ph. D. (Physics and Mathematics), Senior Researcher

11, Surganov Str., 220072, Minsk 

koledad@rambler.ruИнститут математики Национальной академии наук БеларусиБеларусьInstitute of Mathematics of the National Academy of Sciences of BelarusBelarus202629082026704271279Copyright © Коледа Д.В., 20262026Коледа Д.В.Koleda D.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://doklady.belnauka.by/jour/article/view/1318

Рассматривается вопрос о том, как для заданного числового поля K распределены в R и в C алгебраические числа α фиксированной степени deg Kα = n и высоты H Q ( ) α ≤ при Q → ∞. Установлена асимптотика количества таких алгебраических чисел, лежащих во множестве S ⊂ C, в случаях, когда S – интервал вещественной прямой или область комплексной плоскости с липшицевой границей. В частности, показано, что асимптотическая плотность распределения таких алгебраических чисел полностью определяется их степенью над K, внутренним устройством высотной функции H и тем, какое поле – R или C – получается при пополнении поля K по архимедовой метрике.

In the paper we consider the question: given a number field K, what is the distribution of algebraic numbers α of fixed degree deg Kα = n and height H Q ( ) α ≤ in R and in C as Q → ∞. We establish the asymptotic count of such algebraic numbers lying in a set S ⊂ C in cases when S is an interval of the real line or a region of the complex plane with a Lipschitz-parametrizable boundary. In particular, we show that the asymptotic distribution density function of such algebraic numbers is completely determined by their degree over K, by the internal structure of the height function H and by what the completion of K with respect to the archimedean metric is (whether it is R or C ).

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The authors declare that there are no conflicts of interest present.