2021/12/26 by Michaël Bensimhoun, Bensimhoun, Michaël
Mathematics · #Advanced Differential Equations and Dynamical Systems #Analytic and geometric function theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2112.14711
openalex publication_date 2021/12/26 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
Let \mathcal R be a principal ideal domain and \mathcal K = \rm quot(\mathcal R). Assume that P1,… Pn∈ \mathcal K[X] are polynomials which take \mathcal R to \mathcal R, and P is their product. If the Pi satisfy necessary conditions, there exists an infinite set S such that the elements P(m) are mutually coprime as m varies in S, and Pi(m) is coprime to Pj(m) for every i≠ j. We prove that, in addition to the above property, if Pi has no multiple roots whenever i belongs to some subset of \1,… n\, S can be constructed in such a way that Pi(m) is divisible by some prime p, but not by p2. 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.