Irreducibility criteria for pairs of polynomials whose resultant is a prime number
Nicolae Ciprian Bonciocat
TL;DR
The paper develops a broad toolkit for proving irreducibility of pairs of univariate and multivariate polynomials with integer coefficients by exploiting the primality or large-prime-factor structure of their resultant. Central to the approach are root-separation bounds and divisor-based refinements via the quantities d_k and Ω, yielding general theorems that bound the number of irreducible factors and thereby establish irreducibility under suitable distance or coefficient conditions. It specializes to practical cases with g linear or quadratic, connects to primes represented by quadratic forms, and extends to linear combinations Mf+Ng through root-location and Marden-type results. A non-Archimedean, multivariate extension is developed, enabling irreducibility criteria in K[X,Y] and beyond, all illustrated with numerical examples. These results provide concrete, testable irreducibility criteria tied to the algebraic geometry of the resultant and root distribution, with potential applications in number theory and algebraic geometry.
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.