2023/11/30 by Nicolae Ciprian Bonciocat, Bonciocat, Nicolae Ciprian
Computer Science · Mathematics · #11C08 #11R09 #FOS: Mathematics #Mathematical and Theoretical Analysis #Number Theory (math.NT) #Polynomial and algebraic computation #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2311.18568
openalex publication_date 2023/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ResY(f,g), and on the location of their roots in an algebraic closure of K(X), studied in a non-Archimedean setting.