Density of Special Classes of Polynomials with Squarefree Discriminant

TL;DR

This paper calculates the density of monic polynomials over p-adic integers with squarefree discriminant and maximal order properties, providing explicit density formulas for various polynomial subsets.

Contribution

It introduces explicit density computations for polynomials with squarefree discriminants and maximal orders over p-adic integers, extending understanding of polynomial discriminant distributions.

Findings

Derived formulas for the density of polynomials with squarefree discriminant.

Computed densities for polynomials with maximal order properties.

Extended results to various subsets of monic polynomials over p.

Abstract

In this paper, we consider the problem of determining the density of monic polynomials over $\mathbb{Z}_p$ with squarefree discriminant over various subsets of the set of monic polynomials over $\mathbb{Z}_p$ of fixed degree. We compute the density of polynomials in each subset whose discriminant is squarefree, and we compute the density of polynomials $f$ in each subset such that $\mathbb{Z}_p[x]/(f(x))$ is the maximal order of $\mathbb{Q}_p[x]/(f(x))$.