Interpolation by integer-valued polynomials
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.
interpolating integer-valued polynomials
scientific article; zbMATH DE number 1271338
scientific article