Polynomial Closure

AbstractLetDbe 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

Elasticity for integral-valued polynomials

AbstractThe elasticity of a domain is the upper bound of the ratios of lengths of two decompositions in irreducible factors of nonzero nonunit elements. We show that for a large class of Noetherian domains, including any domain contained in a number field (but not a field), the elasticity of the ring of integral-valued polynomials is infinite

Skolem properties, value-functions, and divisorial ideals

AbstractLet D be the ring of integers of a number field K. It is well known that the ring Int(D) = {f ϵ K[X] ¦ f (D) ⊆ D} of integer-valued polynomials on D is a Prüfer domain. Here we study the divisorial ideals of Int(D) and prove in particular that Int(D) has no divisorial prime ideal.We begin with the local case. We show that, if V is a rank-one discrete valuation domain with finite residue field, then the unitary ideals of Int(V) (that is, the ideals containing nonzero constants) are entirely determined by their values on the completion of V. This improves on the Skolem properties which only deal with finitely generated ideals. We then globalize and consider a Dedekind domain D with finite residue fields. We show that a prime ideal of t(D) is invertible if and only if it is divisorial, and also, in the case where the characteristic of D is 0, if and only if it is an upper to zero which is maximal

Polynômes à valeurs entières

Integer-valued polynomialsPolynômes à valeurs entière