Estimating the greatest common divisor of the value of two polynomials

TL;DR

This paper investigates bounds on the greatest common divisor of polynomial values at integers, relating it to the prime factorization and resultants, providing explicit bounds and asymptotic behavior.

Contribution

It introduces new bounds and asymptotic estimates for the minimal difference between the valuation of the resultant and the gcd of polynomial values.

Findings

Least possible value is $ps^2-s$ for $s \\le p$

Asymptotic behavior is $(p-1)s^2$ for large $s$

Provides bounds for the gcd valuation in polynomial pairs

Abstract

Let $p$ be a fixed prime, and let $v(a)$ stand for the exponent of $p$ in the prime factorization of the integer $a$. Let $f$ and $g$ be two monic polynomials with integer coefficients and nonzero resultant $r$. Write $S$ for the maximum of $v(\gcd (f(n), g(n)))$ over all integers $n$. It is known that $S \le v(r)$. We give various lower and upper bounds for the least possible value of $v(r)-S$ provided that a given power $p^s$ divides both $f(n)$ and $g(n)$ for all $n$. In particular, the least possible value is $ps^2-s$ for $s\le p$ and is asymptotically $(p-1)s^2$ for large $s$.