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
References
1. Cassels J. W. S. An introduction to diophantine approximation. Cambridge University Press, 1957. 166 p.
2. Khintchine A. Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen. Mathematische Annalen, 1924, vol. 92, pp. 115–125 (in German). https://doi.org/10.1007/bf01448437
3. Bernik V. I. Application of Hausdorff dimension in the theory of Diophantine approximation. Acta Arithmetica, 1982, vol. 42, no. 3, pp. 219–253 (in Russian). https://doi.org/10.4064/aa-42-3-219-253
4. Beresnevich V. On approximation of real numbers by real algebraic numbers. Acta Arithmetica, 1999, vol. 90, no. 2, pp. 97–112. https://doi.org/10.4064/aa-90-2-97-112
5. Sprindzhuk V. G. Mahler’s problem in metric number theory. Minsk, 1967. 181 p. (in Russian).
6. Beresnevich V. V. A Groshev type theorem for convergence on manifolds. Acta Mathematica Hungarica, 2002, vol. 94, pp. 99–130. https://doi.org/10.1023/a:1015662722298
7. Beresnevich V., Velani S. Number theory meets wireless communications: an introduction for dummies like us. Beresnevich V., ed. Number Theory Meets Wireless Communications. Springer International Publishing, 2020, pp. 1–67. https://doi.org/10.1007/978-3-030-61303-7_1
8. Bernik V. I., Vasiliev D. V., Kudin A. S. On the number of integer polynomials of a given degree and bounded height with a small derivative at the root of the polynomial. Trudy Instituta Matematiki = Proceedings of the Institute of Mathematics, 2014, no. 2, pp. 3–8 (in Russian).
9. Badziahin D. Simultaneous Diophantine approximation to points on the Veronese curve. Available at: https://arxiv.org/abs/2403.17685 (accessed 20 June 2024).
10. Beresnevich V. V., Bernik V. I., Götze F., Zasimovich E. V., Kalosha N. I. Contribution of Jonas Kubilius to the metric theory of Diophantine approximation of dependent variables. Journal of the Belarusian State University. Mathematics and Informatics, 2021, no. 3, pp. 34–50. https://doi.org/10.33581/2520-6508-2021-3-34-50
11. Bernik V. I., Dodson M. M. Metric Diophantine approximation on manifolds. Cambridge Tracts in Mathematics, 1999, no. 137. 172 p. https://doi.org/10.1017/cbo9780511565991
12. Kemesh O. N., Panteleeva Zh. I., Titova A. V. Exact estimates of the measure of small values of integer polynomials Vesnіk Magіleўskaga dzyarzhaўnaga ўnіversіteta іmya A. A. Kulyashova. Seryya V = Bulletin Mogilev State A. Kuleshov University, Seria B, 2021, no. 1 (57), pp. 81–86 (in Russian).
13. Bernik V. I., Vasilyev D. V., Kalosha N. I., Panteleeva Zh. I. 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. Doklady Natsional’noi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 2023, vol. 67, no. 4, pp. 271–278 (in Russian). https://doi.org/10.29235/1561-8323-2023-67-4-271-278
14. Kharin Yu. S., Agievich S. V., Vasilyev D. V., Matveev G. V. Cryptology. Minsk, 2013. 511 p. (in Russian).
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