Polynomial densities and Heilbronn's criterion

TL;DR

This paper investigates the application of Heilbronn's criterion to polynomial-generated number fields, establishing that a positive proportion of Eisenstein polynomials of degree n fail to produce norm-Euclidean fields, especially as prime p grows large.

Contribution

It extends Heilbronn's criterion to a broad class of polynomials and quantifies the density of such polynomials where the criterion applies, showing most Eisenstein polynomials fail to generate norm-Euclidean fields.

Findings

Proportion of polynomials satisfying Heilbronn's criterion tends to 1 as p increases.

A positive density of Eisenstein polynomials of degree n do not generate norm-Euclidean fields.

Lower bounds on the density are explicitly computed and depend on p and n.

Abstract

Heilbronn gave a sufficient condition for a number field with a totally ramified prime to fail to be norm-Euclidean. We say that Heilbronn's criterion applies to a polynomial $f$ if it applies to the number field $K=\mathbb{Q}[x]/(f)$ generated by $f$. Suppose $n\geq 3$ is odd and $p\geq 5$ is prime with $\gcd(p-1,n)=1$. Let ${F}_{p,n}$ denote the collection of monic polynomials $f\in\mathbb{Z}[x]$ of degree $n$ that are Eisenstein at the prime $p$. We order our polynomials by the natural height $\mathrm{Ht}(f)$. Define $\delta_{p,n}(X)$ to be the proportion of polynomials $f\in {F}_{p,n}$ with $\mathrm{Ht}(f)\leq X$ for which Heilbronn's criterion applies. One has $$\liminf_{X\to\infty}\delta_{p,n}(X)\geq \max\left\{\frac{2}{27}\,,\;1-\varepsilon(p)\right\}\,,$$ where $\varepsilon(p)\to 0$ and is effectively computable. In particular, the lower density tends to $1$ as $p\to\infty$ uniformly in $n$. We also give a version of this result where we weaken the condition on $\gcd(p-1,n)$. As a corollary, we show that given an integer $n\geq 2$, a positive proportion of Eisenstein polynomials of degree $n$ fail to generate norm-Euclidean fields.