Around the gcd of the values of two polynomials

TL;DR

This paper explores the relationship between the gcd of polynomial values at integers and the resultant, providing a new mathematical approach to analyze their common divisors.

Contribution

It introduces a novel method using the resultant to study the gcd of polynomial values, offering a new perspective beyond traditional techniques.

Findings

Resultant effectively characterizes the gcd of polynomial values.

The approach simplifies understanding common divisors of polynomial evaluations.

Provides a new tool for algebraic number theory analysis.

Abstract

We propose a mathematical walk around the gcd of the values $A(n)$ and $B(n)$ of two polynomials evaluated at an integer $n$. This is an opportunity to use a very powerful tool: the resultant.