On the distribution of polynomial discriminants: totally real case

TL;DR

This paper investigates how the discriminants of integer polynomials with all real roots are distributed, providing asymptotic counts for polynomials with bounded discriminant as the height grows large.

Contribution

It offers the first asymptotic evaluation of the number of totally real integer polynomials with bounded discriminant in the specified regime.

Findings

Asymptotic formula for the count of polynomials with all roots real and bounded discriminant.

Quantitative description of discriminant distribution in the totally real polynomial class.

Insights into the behavior of polynomial discriminants as polynomial height increases.

Abstract

In the paper we study the distribution of the discriminant $D(P)$ of polynomials $P$ from the class $\mathcal{P}_{n}(Q)$ of all integer polynomials of degree $n$ and height at most $Q$. We evaluate the asymptotic number of polynomials $P\in \mathcal{P}_{n}(Q)$ having all the roots real and satisfying the inequality $|D(P)|\le X$ as $Q\to\infty$ and $X/Q^{2n-2}\to 0$.