Evaluationally coprime linear polynomials

TL;DR

This paper establishes necessary and sufficient conditions for two linear polynomials with integer coefficients to produce coprime values at a positive proportion of integer inputs, advancing understanding of polynomial coprimality in number theory.

Contribution

It provides a complete characterization of when two linear polynomials are evaluationally coprime at a positive density of integers, a novel result in polynomial evaluation theory.

Findings

Characterization of evaluational coprimality conditions

Conditions for positive density of coprime values

Advancement in polynomial coprimality understanding

Abstract

Two polynomials, $f,g \in \mathbb{Z}[x]$ are evaluationally coprime at x if $\gcd(f(x),g(x))=1$. We give necessary and sufficient conditions for two such linear polynomials to have a positive proportion of evaluated coprime values.