Metric theory of diophantine approximation and asymptotic estimates for the number of polynomials with given discriminants divisible by a large power of a prime number
https://doi.org/10.29235/1561-8323-2023-67-4-271-278
Abstract
Discriminants of polynomials characterize the distribution of roots of polynomials in the complex plane. In recent years, for integer polynomials, exact lower-bound estimates have been obtained for the number of polynomials of a given degree and height. The method of obtaining these estimates is based on Minkowski’s theorems in the geometry of numbers and the metric theory of Diophantine approximation. A new method is proposed and allows one to obtain upperbound estimates for the number of polynomials with bounded discriminants in Archimedean and non-Archimedean metrics. The method generalizes the ideas of H. Davenport, B. Volkman, and V. Sprindzuk that allowed them to obtain significant advances in solving Mahler’s problem.
About the Authors
V. I. Bernik
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Bernik Vasiliy I. – D. Sc. (Physics and Mathematics), Professor, Chief Researcher
11, Surganov Str., 220072, Minsk
D. V. Vasilyev
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Vasilyev Denis V. – Ph. D. (Physics and Mathematics)
11, Surganov Str., 220072, Minsk
N. I. Kalosha
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Kalosha Nikolay I. – Ph. D. (Physics and Mathematics)
11, Surganov Str., 220072, Minsk
Zh. I. Panteleeva
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Panteleeva Zhanna I. – Postgraduate Student
11, Surganov Str., 220072, Minsk
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