danДоклады Национальной академии наук БеларусиDoklady of the National Academy of Sciences of Belarus1561-83232524-2431The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-8323-2020-64-1-7-12dan-847Research ArticleМАТЕМАТИКАMATHEMATICSОценки снизу для количества векторов с алгебраическими координатами вблизи гладких поверхностейLower bounds for the number of vectors with algebraic coordinates near smooth surfacesБударинаН. В.BudarinaN. V.

Бударина Наталья Викторовна - доктор физико-математических наук.

A91 К584, Дублин Роуд, Дандолк

Budarina Nataliya V. - D. Sc. (Physics and Mathematics).

A91 К584, Dublin Road, Dundalk

buda77@mail.ruДиккинсонД.DickinsonD.

Диккинсон Детта - кандидат наук.

Мейнут

Dickinson Detta - Ph. D.

Maynooth

detta.dickinson@mu.ieБерникВ. И.BernikV.

Берник Василий Иванович - доктор физико-математических наук, профессор, главный научный сотрудник.

Ул. Сурганова, 11, 220072, Минск

Bernik Vasiliy I. - D. Sc. (Physics and Mathematics), Professor, Chief researcher.

11, Surganov Str., 220072, Minsk

bernik.vasili@mail.ruТехнологический институтHundalk Institute of TechnologyИрландский национальный университет в МейнутеNational University of IrelandИнститут математики, Национальная академия наук БеларусиInstitute of Mathematics, National Academy of Sciences of Belarus202019032020641712Copyright © Бударина Н.В., Диккинсон Д., Берник В.И., 20202020Бударина Н.В., Диккинсон Д., Берник В.И.Budarina N.V., Dickinson D., Bernik 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/847

Аннотация. Пусть z = f(x, y) - некоторая поверхность в трехмерном евклидовом пространстве. Рассмотрим некоторый слой V, точки которого удовлетворяют неравенству |f(x, y) - z| < Q ~Y, где 0 < у < 1 и Q  - достаточно большое натуральное число. В работах Хаксли, Бересневича, Велани было изучено распределение рациональных точек в V. В данной работе изучается распределение точек с алгебраическими сопряженными действительными координатами ᾱ=α1 α2 α3 в V. При некотором c1 = c1(n) получена оценка снизу вида c2 Q n+1-Y для количества алгебраических чисел степени n ≥ 3 и высоты не более c3 Q.

Let z = f(x, y) be a surface in three-dimensional Euclidean space. Consider a neighborhood V of this surface, whose points satisfy the inequality | f(x, y) - z| < Q  -Y, where 0 < у < 1 and Q  is a sufficiently large positive integer. In the works of Huxley, Beresnevich, Velani, the distribution of rational points in V has been started. In this article, we study the distribution of points with real conjugate algebraic coordinates ᾱ = α1α2α3 in V. For some c1 = c1(n), a lower bound is obtained in the form of c2 Q n+1-Y for the number of algebraic numbers of degree n ≥ 3 and of height at most c3 Q.

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