Substitution and Closure of Sets under Integer-Valued Polynomials
✍ Scribed by Sophie Frisch
- Publisher
- Elsevier Science
- Year
- 1996
- Tongue
- English
- Weight
- 462 KB
- Volume
- 56
- Category
- Article
- ISSN
- 0022-314X
No coin nor oath required. For personal study only.
✦ Synopsis
Let R be a domain and K its quotient-field. For a subset S of K, let F R (S) be the set of polynomials f # K[x] with f (S ) R and define the R-closure of S as the set of those t # K for which f (t) # R for all f # F R (S ). The concept of R-closure was introduced by McQuillan (J. Number Theory 39 (1991), 245 250), who gave a description in terms of closure in P-adic topology, when R is a Dedekind ring with finite residue fields. We introduce a toplogy related to, but weaker than P-adic topology, which allows us to treat ideals of infinite index, and derive a characterization of R-closure when R is a Krull ring. This gives us a criterion for F R (S )= F R (T ), where S and T are subsets of K. As a corollary we get a generalization to Krull rings of R. Gilmer's result (J. Number Theory 33 (1989), 95 100) characterizing those subsets S of a Dedekind ring with finite residue fields for which F R (S )=F R (R).