Apollonius circles and the number of irreducible factors of polynomials

TL;DR

This paper establishes upper bounds on the sum of multiplicities of irreducible factors of polynomials with roots inside specific Apollonius circles, extending results to multivariate polynomials over arbitrary fields.

Contribution

It introduces new bounds for polynomial factorization based on geometric root placement within Apollonius circles, including multivariate and non-Archimedean cases.

Findings

Upper bounds for irreducible factor multiplicities in polynomials with roots in Apollonius circles

Results for polynomials with few prime factors at specific evaluation points

Extensions to multivariate and non-Archimedean polynomial settings

Abstract

We provide upper bounds for the sum of the multiplicities of the non-constant irreducible factors that appear in the canonical decomposition of a polynomial $f(X)\in\mathbb{Z}[X]$, in case all the roots of $f$ 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 admissible divisors of $f(a)$ and $f(b)$. In particular, we obtain such upper bounds for the case where $f(a)$ and $f(b)$ have few prime factors, and $f$ is an Enestr\"om-Kakeya polynomial, or a Littlewood polynomial, or has a large leading coefficient. Similar results are also obtained for multivariate polynomials over arbitrary fields, in a non-Archimedean setting.