Frisch, Sophie Interpolation by integer-valued polynomials. (English) Zbl 0927.13023 J. Algebra 211, No. 2, 562-577 (1999). The author pursues two directions to construct interpolating integer-valued polynomials on Krull domains \(R\), that means, given distinct \(a_1, \dots, a_n\in S\leq R\) and \(b_1, \dots, b_n\in R\) there exists an \(f\in \text{Int}(S,R)= \{f\in K[x] \mid f(S)\subseteq R\}\), \(K\) being the quotient field of \(R\), with \(f(a_i) =b_i\), \(i=1, \dots,n\). One approach, running along classical lines, culminates in the following result:An interpolating \(f\in \text{Int} (R,R)\) exists if and only if the \(a_i\) are pairwise incongruent mod all \(P\in\text{Spec}^1(R)\) with \([R:P]= \infty\). The second one is based on so-called weak \(v\)-sequences for \(R\) \((v\) being a valuation of \(R)\) and binomial polynomials constructible from them. Here the corresponding result claims that given an infinite subring \(S\) of \(R\) an \(f\in\text{Int}(S,R)\) exists which interpolates on \(a_1, \dots, a_n\) if this set is a weak \(v\)-sequence for all essential valuations of \(R\). Investigations on the degree of the interpolating polynomial are included. Reviewer: G.Kowol (Wien) Cited in 10 Documents MSC: 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 13B25 Polynomials over commutative rings Keywords:\(v\)-sequence; interpolating integer-valued polynomials; Krull domains × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Amice, Y., Interpolation \(p\), Bull. Soc. Math. France, 92, 117-180 (1964) · Zbl 0158.30201 [2] Cahen, P.-J., Integer-valued polynomials on a subset, Proc. Amer. Math. Soc., 117, 919-929 (1993) · Zbl 0781.13013 [3] Cahen, P.-J., Polynômes à valeurs entières, Canad. J. Math., 24, 747-754 (1972) · Zbl 0224.13006 [4] Cahen, P.-J.; Chabert, J.-L., Integer-Valued Polynomials. Integer-Valued Polynomials, Mathematical Surveys and Monographs, 48 (1997), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0884.13010 [5] Carlitz, L., Finite sums and interpolation formulas over GF[\(p^n},x\), Duke Math. J., 15, 1001-1012 (1948) · Zbl 0032.00303 [6] Chabert, J.-L., Le groupe de Picard de l’anneau des polynômes à valeurs entières, J. Algebra, 150, 213-230 (1992) · Zbl 0776.13008 [7] Frisch, S., Integer-valued polynomials on Krull rings, Proc. Amer. Math. Soc., 124, 3595-3604 (1996) · Zbl 0880.13010 [8] Frisch, S., Binomial coefficients generalized with respect to a discrete valuation, (Bergum, D. E., Applications of Fibonacci numbers. Applications of Fibonacci numbers, Proc. of Graz Conf. 1996, 7 (1998), Kluwer) · Zbl 0977.11009 [9] Gilmer, R.; Heinzer, W.; Lantz, D., The Noetherian property in rings of integer-valued polynomials, Trans. Amer. Math. Soc., 338, 187-199 (1993) · Zbl 0780.13009 [10] Gunji, H.; McQuillan, D. L., On a class of ideals in an algebraic number field, J. Number Theory, 2, 207-222 (1970) · Zbl 0199.37402 [11] Kummer, E. E., Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. Reine Angew. Math., 44, 93-146 (1852) · ERAM 044.1198cj [12] McQuillan, D. L., On Prüfer domains of polynomials, J. Reine Angew. Math., 358, 162-178 (1985) · Zbl 0568.13003 [13] McQuillan, D. L., Split primes and integer-valued polynomials, J. Number Theory, 43, 216-219 (1993) · Zbl 0770.13003 [14] Ostrowski, A., Über ganzwertige Polynome in algebraischen Zahlköpern, J. Reine Angew. Math., 149, 117-124 (1919) · JFM 47.0163.05 [15] Pólya, G., Über ganzwertige Polynome in algebraischen Zahlköpern, J. Reine Angew. Math., 149, 97-116 (1919) · JFM 47.0163.04 [16] Wagner, C. G., Interpolation series for continuous functions on π-adic completions of GF \((qx\), 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.