Integral-valued polynomials over sets of algebraic integers of bounded degree
Let \(R\) be a domain with quotient field \(K\). A subset \(A\) of \(R\) is said to be \textit{polynomially dense} if every polynomial \(f\in K[X]\) satisfying \(f(A)\subset R\) maps \(R\) into \(R\). The author deduces first from a result of \textit{R. Gilmer} [J. Number Theory 33, No. 1, 95--100 (1989; Zbl 0695.13015)] that if \(R\) is an order in the ring of integers in an algebraic number field of degree \(n\), and \(A_n\subset R\) consists of integers of degree \(n\), then \(A_n\) is polynomially dense (Theorem 2.1). Let \({\mathcal A}_n\) be the set of all algebraic integers of degree \(\leq n\), and for \(\alpha\in{\mathcal A}_n\) put \[ R_\alpha =\{f\in Q[X]:\;f(\alpha)\in Z[\alpha]\},\quad S_\alpha =\{f\in Q[X]:\;f(\alpha)\in Z_\alpha\}, \] \(Z_\alpha\) denoting the ring of integers in the field generated by \(\alpha\). It is shown (Theorem 3.1) that for any subset \({\mathcal A}\) of \({\mathcal A}_n\) the integral closure of \[ R_{{\mathcal A}} = \bigcap_{\alpha\in{\mathcal A}}R_\alpha \] equals \[ S_{{\mathcal A}} = \bigcap_{\alpha\in{\mathcal A}}S_\alpha. \] This is used to show (Theorem 1.2) that if \(A_n\) denotes the set of all algebraic integers of degree \(n\), then the sets \(\{f\in Q[X]:\;f({\mathcal A}_n)\subset {\mathcal A}_n\}\) and \(\{f\in Q[X]:\;f(A_n)\subset {\mathcal A}_n\}\) coincide. In the last part of the paper the author considers polynomials \(f\in Q[X]\) mapping \(M_n(Z)\) (the ring of \(n\times n\) matrices with integral entries) or a subset of it into \(M_n(Z)\).
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