Infinite sets of Mutually Coprime Locally Square-free Elements, Inside the Range of Polynomials with Values in Principal Ideal Domains

Abstract

Let $\mathcal R$ be a principal ideal domain and $\mathcal K = {\rm quot}(\mathcal R)$. Assume that $P_1,\ldots P_n\in \mathcal K[X]$ are polynomials which take $\mathcal R$ to $\mathcal R$, and $P$ is their product. If the $P_i$ satisfy necessary conditions, there exists an infinite set $S$ such that the elements $P(m)$ are mutually coprime as $m$ varies in $S$, and $P_i(m)$ is coprime to $P_j(m)$ for every $i\ne j$. We prove that, in addition to the above property, if $P_i$ has no multiple roots whenever $i$ belongs to some subset of $\{1,\ldots n\}$, $S$ can be constructed in such a way that $P_i(m)$ is divisible by some prime $p$, but not by $p^2$. This result is the basis of a conjecture formulated at the end of this article, according to which one can extract infinitely many square-free elements from the value set of $P$, provided $P$ has no multiple root. In the course of this article, the notion of totally primitive polynomial is introduced and elaborated in order to provide a convenient framework for those questions, or other questions concerning the value set of polynomials. This framework makes possible more enlightening, and seemingly more general, statements of the famous Bunyakovsky and Schinzel conjectures.