Apollonius circles and irreducibility criteria for polynomials

TL;DR

This paper establishes new irreducibility criteria for integer polynomials based on the geometric placement of roots within Apollonius circles, linking root location to polynomial factorization properties.

Contribution

It introduces novel irreducibility criteria involving Apollonius circles and extends results to multivariate polynomials over arbitrary fields in non-Archimedean contexts.

Findings

Proves irreducibility of polynomials with roots inside specific Apollonius circles.

Provides criteria for polynomials with few prime factors at specific points.

Extends irreducibility results to multivariate and non-Archimedean settings.

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 abscisae $a$ 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 Enestr\"om-Kakeya polynomial, or has a large leading coefficient. Analogous results are also provided for multivariate polynomials over arbitrary fields, in a non-Archimedean setting.