TY - JOUR
T1 - Generalized Factorials and Fixed Divisors over Subsets of a Dedekind Domain
AU - Bhargava, Manjul
N1 - Funding Information:
This work was done during the 1995 Summer Research Program at the University of Minnesota, Duluth, directed by Joseph Gallian and sponsored by the National Science Foundation (Grant DMS-9225045) and the National Security Agency (Grant number MDA 904-91-H-0036).
PY - 1998/9
Y1 - 1998/9
N2 - Given a subsetXof a Dedekind domainD, and a polynomialF∈D[x], thefixed divisor d(X, F) ofFoverXis defined to be the ideal inDgenerated by the elementsF(a),a∈X. 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=Z, this generalized factorial reduces to the ordinary factorial function; hence we obtain as special cases classic results of Pólya, Gunji, and McQuillan relatingd(Z, F) and the usual factorial function.
AB - Given a subsetXof a Dedekind domainD, and a polynomialF∈D[x], thefixed divisor d(X, F) ofFoverXis defined to be the ideal inDgenerated by the elementsF(a),a∈X. 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=Z, this generalized factorial reduces to the ordinary factorial function; hence we obtain as special cases classic results of Pólya, Gunji, and McQuillan relatingd(Z, F) and the usual factorial function.
UR - http://www.scopus.com/inward/record.url?scp=0002588611&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0002588611&partnerID=8YFLogxK
U2 - 10.1006/jnth.1998.2220
DO - 10.1006/jnth.1998.2220
M3 - Article
AN - SCOPUS:0002588611
SN - 0022-314X
VL - 72
SP - 67
EP - 75
JO - Journal of Number Theory
JF - Journal of Number Theory
IS - 1
ER -