2021/05/28 by Arnaud Bodin, Bodin, Arnaud, Pierre Dèbes +1
Mathematics · #11A05 #12E05 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2105.13883
openalex publication_date 2021/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given two polynomials P(\underline x), Q(\underline x) in one or more variables and with integer coefficients, how does the property that they are coprime relate to their values P(\underline n), Q(\underline n) at integer points \underline n being coprime? We show that the set of all gcd (P(\underline n), Q(\underline n)) is stable under gcd and under lcm. A notable consequence is a result of Schinzel: if in addition P and Q have no fixed prime divisor (i.e., no prime dividing all values P(\underline n), Q(\underline n)), then P and Q assume coprime values at "many" integer points. Conversely we show that if "sufficiently many" integer points yield values that are coprime (or of small gcd) then the original polynomials must be coprime. Another noteworthy consequence of this paper is a version over the ring of integers of Hilbert's irreducibility theorem.