  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.
