On the Natural Density of Monic Integer Polynomials with Roots in a Fixed Number Field

TL;DR

This paper analyzes how the density of monic integer polynomials with roots in a fixed number field diminishes as polynomial height increases, revealing a degree-dependent phase transition in decay rates.

Contribution

It provides explicit asymptotic bounds and reveals a phase transition in the decay rate of polynomial densities based on degree, integrating arithmetic and geometric methods.

Findings

Density decays at rate O(H^{-1} log H) for degree 2

Higher degrees exhibit decay rate O(H^{-1}) due to rational roots

Explicit bounds for counting reducible and irreducible polynomials are established

Abstract

In this article, we investigate the statistical distribution and asymptotic behavior of the family of monic integer polynomials of degree $n$ having at least one root in a fixed number field $K$. Although the framework of thin sets implies that the natural density of this family in the parameter space of bounded height is zero, explicitly quantifying this vanishing rate is a central challenge in arithmetic statistics. Employing a hybrid approach that integrates the Mahler measure, Dirichlet's unit theorem, and residue analysis of the Dedekind zeta function, we demonstrate that the rate of convergence of this density to zero is strictly dependent on the degree $n$. Specifically, we prove that the degrees of the factors induce a phase transition in the asymptotic behavior; for polynomials of degree $n = 2$, the decay rate is bounded by $O(H^{-1} \log H)$, whereas for higher degrees, the asymptotic behavior is dominated by the contribution of rational roots, yielding a bound of $O(H^{-1})$. Beyond deriving these asymptotic estimates, we apply principles from the geometry of numbers to establish explicit combinatorial bounds for counting both the reducible and irreducible components of these polynomials. These explicit bounds provide practical tools for computational evaluations within this domain.