On the Krull dimension of domains of integer-valued polynomials
Let \(A\) be an integral domain with quotient field \(K\), and let \(\text{Int}(A)=\{f\in K[X]| f(A)\subset A\}\) be the ring of integer valued polynomials for \(A\). This paper considers the problem of estimating the Krull dimension of \(\text{Int}(A)\) in terms of that of \(A\) and \(A[X]\). It is known for many special classes of \(A\), and conjectured in general, that \(\text{dim}(\text{Int}(A))\leq \dim(A[X])\) (this would also bound \(\text{dim}(\text{Int}(A))\) in terms of \(\text{dim}(A)\) since \textit{A. Seidenberg} showed in 1953 [Pac. J. Math. 3, 505-512 (1953; Zbl 0052.26902)] that \(\dim(A[X])\leq 1+2\dim (A)\)). To investigate this question the authors consider, for \(A\subseteq B\) an extension of integral domains, \(I(B,A)=\{f\in B[X]| f(A)\subseteq A\}\). They show that \(\dim(I(B,A))=\dim(A[X])\) if \(A[X]\) is a Jaffard domain, but give an example in which \(\text{dim}(I(B,A))> \dim(A[X])\). They extend a theorem of Seidenberg on \(A[X]\) to show that for any \(n\), \(h\) such that \(n+1\leq h\leq 2n+1\) there exists a domain \(D\) such that \(\dim(D)=n\) and \(\dim(\text{Int}(D))=h\). The paper concludes with some estimates of the Krull dimension of \(\text{Int}(E,A)=\{f\in K[X]\mid f(E)\subset A\}\) (where \(E\) is a subset of \(K\)) for special classes of domains \(A\).
scientific article; zbMATH DE number 1121251