Generalized factorials and fixed divisors over subsets of a Dedekind domain

Given a subset X of a Dedekind domain D, and a polynomial F is an element of D[x], the fixed divisor d(X, F) of F over X is defined to be the ideal in D generated by the elements F(a), a is an element of X. In this paper we derive a simple expression for d(X, F) explicitly in terms of the coefficients of F, using a generalized notion of "factorial" introduced by the author in a previous paper. When X = D = Z, this generalized factorial reduces to the ordinary factorial function; hence we obtain as special cases classic results of Polya, Gunji, and McQuillan relating d(Z, F) and the usual factorial function. (C) 1998 Academic Press.