Fixed Divisor of a Multivariate Polynomial and Generalized Factorials in Several Variables

TL;DR

This paper introduces new generalized factorials in multiple variables over subsets of Dedekind domains and explores their relationship with the fixed divisor of multivariate polynomials, extending classical results and providing explicit formulas.

Contribution

It generalizes fixed divisor concepts to multivariate polynomials over Dedekind domains using novel factorial definitions and relates fixed divisors to polynomial images and coefficients.

Findings

Generalized factorials in several variables are defined over subsets of Dedekind domains.

Fixed divisor of multivariate polynomials is characterized in terms of polynomial images.

Explicit formulas relate fixed divisors to polynomial coefficients under chosen bases.

Abstract

We define new generalized factorials in several variables over an arbitrary subset $\underline{S} \subseteq R^n,$ where $R$ is a Dedekind domain and $n$ is a positive integer. We then study the properties of the fixed divisor $d(\underline{S},f)$ of a multivariate polynomial $f \in R[x_1,x_2, \ldots, x_n]$. We generalize the results of Polya, Bhargava, Gunji & McQuillan and strengthen that of Evrard, all of which relate the fixed divisor to generalized factorials of $\underline{S}$. We also express $d(\underline{S},f)$ in terms of the images $f(\underline{a})$ of finitely many elements $\underline{a} \in R^n$, generalizing a result of Hensel, and in terms of the coefficients of $f$ under explicit bases.