Generalized Factorials and Fixed Divisors over Subsets of a Dedekind Domain Original Research Article

Given a subsetXof a Dedekind domainD, and a polynomialFset membership, variantD[x], thefixed divisor d(X, F) ofFoverXis defined to be the ideal inDgenerated by the elementsF(a),aset membership, variantX. In this paper we derive a simple expression ford(X, F) explicitly in terms of the coefficients ofF, using a generalized notion of “factorial” introduced by the author in a previous paper. WhenX=D=image, this generalized factorial reduces to the ordinary factorial function; hence we obtain as special cases classic results of Pólya, Gunji, and McQuillan relatingd(image, F) and the usual factorial function.