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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