Lengths of the intervals where integer polynomials can attain small values
https://doi.org/10.29235/1561-8323-2024-68-6-447-453
Abstract
The concept of the discriminant of a quadratic polynomial allows for easy extraction of information about its real and complex roots. The discriminant of a polynomial of an arbitrary degree is also an important characteristic of the polynomial, which proves useful in many problems in the theory of Diophantine approximation. In 2023, Belarusian mathematician D. Badziahin solved a problem posed by Davenport in the 1960s concerning the range of values of discriminants in the cubic case. The paper provides a complete solution to the problem of divisibility of discriminants by large powers of prime numbers in the case of cubic polynomials.
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
A. S. Kudin
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Kudin Alexey S. – Ph. D. (Physics and Mathematics)
11, Surganov Str., 220072, Minsk
Zh. I. Panteleeva
Belarusian State Agrarian Technical University
Belarus
Belarus
Panteleeva Zhanna I. – Senior Lecturer
99, Nezavisimosti Ave., 220012, Minsk
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