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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)].

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13G05 Integral domains

Citations:

Zbl 0887.13010
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References:

[1] Cahen, P.-J.; Chabert, J.-L., Integer-valued polynomials, Amer. Math. Soc. Surveys and Monographs, 48 (1997) · Zbl 0884.13010
[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
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