zbMATH — the first resource for mathematics

Interpolation domains. (English) Zbl 0990.13014
Authors’ abstract: Call a domain \(D\) with quotient field \(K\) an interpolation domain if, for each choice of distinct arguments \(a_1, \dots, a_n\) and arbitrary values \(c_1,\dots, c_n\) in \(D\), there exists an integer-valued polynomial \(f\) (that is, \(f\in K[X]\) with \(f(D)\subseteq (D))\), such that \(f(a_i)= c_i\) for \(1\leq i\leq n\). 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, locally unibranched with finite residue fields, in the second one, if and only if the ring \(\text{Int} (D)=\{f\in K[X] \mid f(D) \subseteq D\}\) of integer-valued polynomials is itself a Prüfer domain. We also show that an interpolation domain must satisfy a double-boundedness condition, and thereby simplify a recent characterization of the domains \(D\) such that \(\text{Int}(D)\) is a Prüfer domain [see K. A. Loper, Proc. Am. Math. Soc. 126, No. 3, 657-660 (1998; Zbl 0887.13010)].

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13G05 Integral domains
Zbl 0887.13010
Full Text: DOI Link
[1] Cahen, P.-J.; Chabert, J.-L., Integer-valued polynomials, Amer. math. soc. surveys and monographs, 48, (1997)
[2] Cahen, P.-J.; Haouat, Y., Polynômes à valeurs entières sur un anneau de pseudo-valuation, Manuscripta math., 61, 23-31, (1988) · Zbl 0656.13004
[3] Chabert, J.-L., Integer-valued polynomials, Prüfer domains and localization, Proc. amer. math. soc., 118, 1061-1073, (1993) · Zbl 0781.13014
[4] Carlitz, L., Finite sums and interpolation formulas over GF[p^{n},x], Duke math. J., 15, 1001-1012, (1948) · Zbl 0032.00303
[5] Frisch, S., Interpolation by integer-valued polynomials, J. algebra, 211, 562-577, (1999) · Zbl 0927.13023
[6] Loper, A., A classification of all domains D such that int(D) is a Prüfer domain, Proc. amer. math. soc., 126, 657-660, (1998) · Zbl 0887.13010
[7] Wagner, C.G., Interpolation series for continuous functions on π-adic completions of GF(q,x), Acta arith., 17, 389-406, (1971) · Zbl 0223.12009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.