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

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 $f\in R[x_1,x_2,\dots ,x_n]$. 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.

Accepted for publication in J. Korean Math. Soc