Polynomial densities and Heilbronn's criterion
Alexis Hibbler, Kevin J. McGown, Enrique Treviño
TL;DR
This work analyzes the frequency with which Heilbronn's criterion applies to polynomials f that are Eisenstein at a prime p, thereby implying that the associated number field K=Q[x]/(f) is not norm-Euclidean. Using a sieve-like framework with local densities, the authors derive a lower bound on the proportion of p-Eisenstein polynomials for which Heilbronn's criterion holds, and show this proportion approaches 1 as p grows, uniformly in odd n≥3 under gcd(p−1,n)=1 (with a version for relaxed gcd conditions). A key outcome is that a positive proportion of Eisenstein polynomials of any fixed degree n≥2 fail to generate norm-Euclidean fields. The results leverage explicit local density computations, no-root conditions modulo primes, and controlled representations of p as linear combinations of small primes to invoke Heilbronn's criterion.
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 $\mathcal{F}_{p,n}$ 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 $δ_{p,n}(X)$ to be the proportion of polynomials $f\in\mathcal{F}_{p,n}$ with $\mathrm{Ht}(f)\leq X$ for which Heilbronn's criterion applies. One has $$\liminf_{X\to\infty}δ_{p,n}(X)\geq \max\left\{\frac{2}{27}\,,\;1-\varepsilon(p)\right\}\,,$$ 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.