2011/09/19 by Jesse Elliott, Elliott, Jesse
Computer Science · Mathematics · #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Rings, Modules, and Algebras #math.AC
paper · pdf · doi:10.48550/arxiv.1109.3921
21 pages
arxiv created 2011/09/19 · openalex publication_date 2011/09/19 · arxiv updated 2011/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let D be an integral domain with quotient field K. For any set X, the ring Int(DX) of integer-valued polynomials on DX is the set of all polynomials f in K[X] such that f(DX) is a subset of D. Using the t-closure operation on fractional ideals, we find for any set X a D-algebra presentation of Int(DX) by generators and relations for a large class of domains D, including any unique factorization domain D, and more generally any Krull domain D such that Int(D) has a regular basis, that is, a D-module basis consisting of exactly one polynomial of each degree. As a corollary we find for all such domains D an intrinsic characterization of the D-algebras that are isomorphic to a quotient of Int(DX) for some set X. We also generalize the well-known result that a Krull domain D has a regular basis if and only if the Polya-Ostrowski group of D (that is, the subgroup of the class group of D generated by the images of the factorial ideals of D) is trivial, if and only if the product of the height one prime ideals of finite norm q is principal for every q.