title: Integral-valued polynomials over sets of algebraic integers of bounded degree.
abstract: LetKbe a number field of degreenwith ring of integers [Formula: see text]. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if [Formula: see text] maps every element of [Formula: see text] of degreento an algebraic integer, then [Formula: see text] is integral-valued over [Formula: see text], that is, [Formula: see text]. A similar property holds if we consider the set of all algebraic integers of degreenand a polynomial [Formula: see text]: if [Formula: see text] is integral over [Formula: see text] for every algebraic integerαof degreen, then [Formula: see text] is integral over [Formula: see text] for every algebraic integerβof degree smaller thann. This second result is established by proving that the integral closure of the ring of polynomials in [Formula: see text] which are integer-valued over the set of matrices [Formula: see text] is equal to the ring of integral-valued polynomials over the set of algebraic integers of degree equal ton.
authors.affiliation: Institut für Analysis und Comput. Number Theory, Technische Universität, Steyrergasse 30, A-8010 Graz, Austria.
authors.forename: Giulio
authors.initials: G
authors.lastname: Peruginelli
keywords: 11C; 13B25; 13F20; Algebraic integers with bounded degree; Integer-valued polynomial; Integral closure; Polynomially dense subset; Prüfer domain; Pullback