The Density of Shifted and Affine Eisenstein Polynomials

TL;DR

This paper determines the proportion of polynomials that become Eisenstein irreducible after a shift, using a local-to-global density approach, thus answering a question about polynomial irreducibility criteria.

Contribution

It provides a complete density characterization for shifted Eisenstein polynomials, employing a novel local-to-global method for density calculations.

Findings

Established the exact natural density of shifted Eisenstein polynomials.

Applied a local-to-global principle to compute densities over free $\

Confirmed the density results for affine transformations of polynomials.

Abstract

In this paper we provide a complete answer to a question by Heyman and Shparlinski concerning the natural density of polynomials which are irreducible by Eisenstein's criterion after applying some shift. The main tool we use is a local to global principle for density computations over a free $\mathbb Z$-module of finite rank.