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We define new generalized factorials in several variables over an arbitrarysubset $\underline{S} \subseteq R^n,$ where $R$ is a Dedekind domain and $n$ isa 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 andstrengthen that of Evrard, all of which relate the fixed divisor to generalizedfactorials of $\underline{S}$. We also express $d(\underline{S},f)$ in terms ofthe images $f(\underline{a})$ of finitely many elements $\underline{a} \inR^n$, generalizing a result of Hensel, and in terms of the coefficients of $f$under explicit bases.

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