Integral-valued polynomials over the set of algebraic integers of bounded degree
Giulio Peruginelli
备注：
keywords: Integer-valued polynomial, Algebraic integers with bounded degree, Prüfer domain, Polynomially dense subset, Integral closure. To appear in Journal of Number Theory
Let $K$ be a number field of degree $n$ with ring of integers $O_K$. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if $h\in K[X]$ maps every element of $O_K$ of degree $n$ to an algebraic integer, then $h(X)$ is integral-valued over $O_K$, that is $h(O_K)\subset O_K$. A similar property holds if we consider the set of all algebraic integers of degree $n$ and a polynomial $f\in\mathbb{Q}[X]$: if $f(α)$ is integral over $\mathbb{Z}$ for every algebraic integer $α$ of degree $n$, then $f(β)$ is integral over $\mathbb{Z}$ for every algebraic integer $β$ of degree smaller than $n$. This second result is established by proving that the integral closure of the ring of polynomials in $\mathbb{Q}[X]$ which are integer-valued over the set of matrices $M_n(\mathbb{Z})$ is equal to the ring of integral-valued polynomials over the set of algebraic integers of degree equal to $n$.
The paper studies some properties of the ring of integer-valued quasi-polynomials. On this ring, theory of generalized Euclidean division and generalized GCD are presented. Applications to finite simple continued fraction expansion and Smith normal form of integral matrices with integer parameters are also given.
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It is well known, that if polynomial with rational coefficients of degree $n$ takes integer values in points $0,1,...,n$ then it takes integer values in all integer points. Are there sets of $n+1$ points with the same property in other integral domains? We show that answer is negative for the ring of Gaussian integers $\mathbb{Z}[i]$ when $n$ is large enough. Also we discuss the question about minimal possible size of set, such that if polynomial takes integer values in all points of this set then it is integer-valued.
An integer--valued function is an entire function which maps the nonnegative integers $\mathbb N$ to the integers. An example is $2^z$. A Hurwitz function is an entire function having all derivatives taking integer values at $0$. An example is ${\mathrm e}^z$. Lower bound for the growth order of such functions have a rich history. Many variants have been considered: for instance, assuming that the first $k$ derivatives at the integers are integers, or assuming that the derivatives at $k$ points are integers. These as well as and many other variants have been considered. We survey some of them.
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