The asymptotic number of integral cubic polynomials with bounded heights and discriminants

TL;DR

This paper derives an asymptotic formula for counting integral cubic polynomials with bounded height and discriminant, advancing understanding of their distribution and providing optimal estimates within a specific discriminant range.

Contribution

It presents the first asymptotic formula for the number of integral cubic polynomials with bounded height and discriminant, improving previous bounds and establishing the main term as optimal.

Findings

Established the asymptotic count for such polynomials

Proved the main term is the best possible within the considered range

Partially solved the distribution problem for cubic polynomial discriminants

Abstract

In the paper we partially solved the problem of the distribution of the discriminants of integral polynomials in the cubic case. We proved the asymptotic formula for the number of integral cubic polynomials having bounded height and bounded discriminant. The main term of the obtained formula is the best possible for the considered range of polynomial discriminant values.