Discriminants of Fields Generated by Polynomials of Given Height

TL;DR

This paper provides bounds on the number of irreducible polynomials of fixed degree and height with roots generating fields of specified discriminants, advancing understanding of discriminant distribution among polynomial-generated fields.

Contribution

It introduces new upper bounds for the count of such polynomials and improves existing bounds for trinomials with given discriminant square-free parts.

Findings

Established upper bounds for polynomials with specified discriminants.

Derived a lower bound on the number of distinct discriminants for degree n polynomials.

Improved bounds for the count of trinomials with given discriminant properties.

Abstract

We obtain upper bounds for the number of monic irreducible polynomials over $\mathbb Z$ of a fixed degree $n$ and a growing height $H$ for which the field generated by one of its roots has a given discriminant. We approach it via counting square-free parts of polynomial discriminants via two complementing approaches. In turn, this leads to a lower bound on the number of distinct discriminants of fields generated by roots of polynomials of degree $n$ and height at most $H$. We also give an upper bound for the number of trinomials of bounded height with given square-free part of the discriminant, improving previous results of I. E. Shparlinski (2010).