Integral-valued polynomials over the set of algebraic integers of bounded degree

TL;DR

This paper investigates conditions under which polynomials map algebraic integers of bounded degree to algebraic integers, establishing new criteria for integral-valued polynomials over algebraic integers of fixed degree.

Contribution

It provides novel criteria linking polynomial mappings to algebraic integers of bounded degree, extending understanding of integral-valued polynomials over number fields.

Findings

Polynomials mapping degree n algebraic integers to algebraic integers are integral-valued over the ring of integers.

If a polynomial maps degree n algebraic integers to algebraic integers, it also does so for all smaller degrees.

The integral closure of polynomials integer-valued over matrices equals those integer-valued over algebraic integers of degree n.

Abstract

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 $f(\alpha)$ is integral over $\mathbb{Z}$ for every algebraic integer $\alpha$ of degree $n$, then $f(\beta)$ is integral over $\mathbb{Z}$ for every algebraic integer $\beta$ 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 $n$.