Estimating the greatest common divisor of the value of two polynomials

Abstract

Let [Formula: see text] be a fixed prime, and let [Formula: see text] stand for the exponent of [Formula: see text] in the prime factorization of the integer [Formula: see text]. Let [Formula: see text] and [Formula: see text] be two monic polynomials with integer coefficients and nonzero resultant [Formula: see text]. Write [Formula: see text] for the maximum of [Formula: see text] over all integers [Formula: see text]. It is known that [Formula: see text]. We give various lower and upper bounds for the least possible value of [Formula: see text] provided that a given power [Formula: see text] divides both [Formula: see text] and [Formula: see text] for all [Formula: see text]. In particular, the least possible value is [Formula: see text] for [Formula: see text] and is asymptotically [Formula: see text] for large [Formula: see text].