with P.-J. Cahen and J.-L. Chabert, J. Algebra 225 (2000) 794-803

Interpolation Domains

Abstract: Call a domain D an interpolation domain if, for each finite set (a_1,...,a_n) of distinct elements of D and each corresponding set of ``values'' (c_1,...,c_n) in D, there exists f in Int(D) such that f(a_i) = c_i for 1 <= i<= n. (Here Int(D) denotes the ring of integer-valued polynomials on D, i.e. the ring of polynomials f with coefficients in the quotient field of D with the property that f(d) is in D for every d in D.)

We characterize completely the interpolation domains if D is Noetherian or a Prüfer domain. In the first case, we show that D is an interpolation domain if and only if it is one-dimensional and locally unibranched with finite residue fields; in the second case, if and only if the ring Int(D) of integer-valued polynomials is itself a Prüfer domain.

In general, we show that an interpolation domain must satisfy a double boundedness condition; thus we generalize and simplify Loper's recent characterization of the domains D such that the ring Int(D) of integer-valued polynomials is a Prüfer domain.

1991 Mathematics Subject Classification:
Primary 13B25, 13F05, 13F20; Secondary 13B22, 13B30, 13F30.

PDF: P.-J. Cahen, J.-L. Chabert, S. Frisch, Interpolation domains
DVI: Cahen, Chabert, Frisch, Interpolation domains (dvi file)
PS: Cahen, Chabert, Frisch, Interpolation domains (PostScript)

back to Sophie Frisch's homepage