arXiv subjects

Michaël Bensimhoun

Publications and source records attributed to Michaël Bensimhoun.

3 recordsLinked to original sources

arXiv2024

A Framework for Asymptotic Limit Problems of Probabilistic Nature

A convenient framework for dealing with asymptotic limit problems of probabilistic nature is provided. These problems include questions such as finding the asymptotic proportion of terms of a sequence falling inside a given interval, or the limit of the arithmetic mean of its partial sums; but several classes of problems are examined in a much more general setting. The proposed framework, which aims to unify those questions and their solution, is based on the idea that to any finite multiset $E_n$, one can associate a finitely distributed atomic probability $μ_n$; assuming $μ_n$ tends in distribution to a probability $μ$, it provides the tools needed to establish the desired asymptotic limit. Few examples are worked out in order to illustrate how using the framework.

math.HO↗

arXiv2022

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

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.

math.NT↗

arXiv2014

A note on the product of the conjugates of a polynomial

The theorem proved in this note, although elementary, is related to a certain misconception. If $K$ is a field, $f\in K[X]$ is separable and irreducible over $K$, and $g$ is a polynomial dividing $f$, whose coefficients lie in some finite Galois extension of $K$, it may seem natural to assert that the product of the conjugates of $g$ over $K[X]$ is $f$. But this assertion is wrong, except in one particular case. In this note, we make the relation between $K$, $f$, the product of the conjugates of $g$, and the coefficient field of $g$, precise. In particular, it is shown that the product of the conjugates of $g$ over $K[X]$ is equal to $f^n$, with $n\in \mathbb N$.

math.GM↗