On the Number of Eisenstein Polynomials of Bounded Height

TL;DR

This paper refines the asymptotic count of monic Eisenstein polynomials of fixed degree and bounded height, providing explicit error bounds and extending results to all Eisenstein polynomials.

Contribution

It offers a more precise asymptotic formula with explicit error bounds for counting Eisenstein polynomials of bounded height, extending prior work to non-monic cases.

Findings

Explicit asymptotic formula for monic Eisenstein polynomials

Error bounds for the asymptotic approximation

Asymptotic count for all Eisenstein polynomials

Abstract

We obtain a more precise version of an asymptotic formula of A. Dubickas for the number of monic Eisenstein polynomials of fixed degree $d$ and of height at most $H$, as $H\to \infty$. In particular, we give an explicit bound for the error term. We also obtain an asymptotic formula for arbitrary Eisenstein polynomials of height at most $H$.