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We provide upper bounds for the sum of the multiplicities of the non-constantirreducible 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 Apolloniuscircle 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 admissibledivisors of $f(a)$ and $f(b)$. In particular, we obtain such upper bounds forthe case where $f(a)$ and $f(b)$ have few prime factors, and $f$ is anEnestr\"om-Kakeya polynomial, or a Littlewood polynomial, or has a largeleading coefficient. Similar results are also obtained for multivariatepolynomials over arbitrary fields, in a non-Archimedean setting.

Apollonius circles and the number of irreducible factors of polynomials