On Shifted Eisenstein Polynomials

TL;DR

This paper investigates polynomials that become Eisenstein polynomials after a variable shift, establishing bounds on the shift, density comparisons with classical Eisenstein polynomials, and the infinitude of certain irreducible polynomials.

Contribution

It introduces the concept of shifted Eisenstein polynomials, provides bounds on the necessary shift, and compares their density to classical Eisenstein polynomials.

Findings

Upper bound on the shift needed for shifted Eisenstein polynomials

Lower bound on the density of shifted Eisenstein polynomials

Infinitude of irreducible polynomials not classified as shifted Eisenstein

Abstract

We study polynomials with integer coefficients which become Eisenstein polynomials after the additive shift of a variable. We call such polynomials shifted Eisenstein polynomials. We determine an upper bound on the maximum shift that is needed given a shifted Eisenstein polynomial and also provide a lower bound on the density of shifted Eisenstein polynomials, which is strictly greater than the density of classical Eisenstein polynomials. We also show that the number of irreducible degree $n$ polynomials that are not shifted Eisenstein polynomials is infinite. We conclude with some numerical results on the densities of shifted Eisenstein polynomials.