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
![$f(X)\in\mathbb{Z}[X]$](img1.png)
, in case all
the roots of
![$f$](img2.png)
lie inside an Apollonius circle associated to two
points on the real axis with integer abscissae
![$a$](img3.png)
and
![$b$](img4.png)
, with
ratio of the distances to these points depending on the admissible
divisors of
![$f(a)$](img5.png)
and
![$f(b)$](img6.png)
. In particular, we obtain such upper
bounds for the case where
![$f(a)$](img5.png)
and
![$f(b)$](img6.png)
have few prime factors,
and
![$f$](img2.png)
is an Eneström-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.