Irreducibility criteria for pairs of polynomials whose resultant is a prime number

Abstract

We obtain various irreducibility criteria for pairs of polynomials $(f(X),g(X))$ with integer coefficients whose resultant $Res(f,g)$ is a prime number, or is divisible by a sufficiently large prime number, and also for some of their linear combinations $Mf(X)+Ng(X)$ with integer scalars $M$ and $N$. In particular, we find irreducibility conditions for polynomials with coefficients obtained by representing primes by certain quadratic forms. The irreducibility criteria will appear as corollaries of more general results providing upper bounds for the number of irreducible factors of each one of $f$ and $g$, counting multiplicities, that depend on the prime factorization of $Res(f,g)$, and on the distances between the roots of $f$ and those of $g$. Similar results will be also obtained for pairs of bivariate polynomials $(f(X,Y),g(X,Y))$ over an arbitrary field $K$, using information on the canonical decomposition of their resultant $Res_Y(f,g)$, and on the location of their roots in an algebraic closure of $K(X)$, studied in a non-Archimedean setting.