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.
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More From: Doklady of the National Academy of Sciences of Belarus
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