On the greatest common divisor of the value of two polynomials

TL;DR

This paper demonstrates that for two monic integer polynomials with a square-free resultant, every positive divisor of that resultant can be realized as the gcd of their values at some integer point.

Contribution

It establishes a novel connection between the divisors of the resultant and the gcd of polynomial values at integers under the square-free condition.

Findings

All positive divisors of the resultant are attainable as gcd values.

The result applies specifically to monic polynomials with square-free resultants.

Provides a new perspective on the relationship between resultants and polynomial evaluations.

Abstract

We show that if two monic polynomials with integer coefficients have square-free resultant, then all positive divisors of the resultant arise as the greatest common divisor of the values of the two polynomials at a suitable integer.