J. Number Theory, 56 (1996) 396-403 [MR 97b:13011; Zb 844.13014]

Substitution and Closure of Sets under Integer-Valued Polynomials

Abstract: We chracterize polynomial closure of subsets of Krull rings. 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 in K[x] which, when acting as a function on K by substitution of the variable, map S into R. The polynomial closure of S is the set of those t in K for which f(t) in R for all f in F_R(S). The concept of polynomial closure was (under the name of R-closure) 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 polynomial 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 R with finite residue fields for which F_R(S)=F_R(R).

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