Polynomial Closure

Abstract

LetDbe a domain with quotient fieldK. The polynomial closure of a subsetEofKis the largest subsetFofKsuch that each polynomial (with coefficients inK), which mapsEintoD, maps alsoFintoD. In this paper we show that the closure of a fractional ideal is a fractional ideal, that divisorial ideals are closed and that conversely closed ideals are divisorial for a Krull domain. IfDis a Zariski ring, the polynomial closure of a subset is shown to contain its topological closure; the two closures are the same ifDis a one-dimensional Notherian local domain, with finite residue field, which is analytically irreducible. A subset ofDis said to be polynomially dense inDif its polynomial closure isDitself. The characterization of such subsets is applied to determine the ringRαformed by the valuesf(α) of the integer-valued polynomials on a Dedekind domainR(at some elementαof an extension ofR). It is also applied to generalize a characterization of the Noetherian domainsDsuch that the ring Int(D) of integer-valued polynomials onDis contained in the ring Int(D′) of integer-valued polynomials on the integral closureD′ ofD.