Apollonius circles and irreducibility criteria for polynomials

Abstract

We prove the irreducibility of integer polynomials f(X) whose roots lie inside an Apollonius circle associated to two points on the real axis with integer abscissae a and b, with ratio of the distances to these points depending on the canonical decomposition of f(a) and f(b). In particular, we obtain irreducibility criteria for the case where f(a) and f(b) have few prime factors, and f is either an Eneström–Kakeya polynomial, or has a large leading coefficient. Analogous results are also provided for multivariate polynomials over arbitrary fields, in a non-Archimedean setting.