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arXiv · 2303.04033

Apollonius circles and the number of irreducible factors of polynomials

Anca Iuliana Bonciocat·Nicolae Ciprian Bonciocat·Yann Bugeaud·Mihai Cipu·Maurice Mignotte
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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.

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Anca Iuliana Bonciocat, Nicolae Ciprian Bonciocat, Yann Bugeaud, Mihai Cipu, Maurice Mignotte. 2023-03-07. Apollonius circles and the number of irreducible factors of polynomials. https://arxiv.org/abs/2303.04033

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