Integer-valued polynomials over matrices and divided differences

Abstract

Let $D$ be an integrally closed domain with quotient field $K$ and $n$ a positive integer. We give a characterization of the polynomials in $K[X]$ which are integer-valued over the set of matrices $M_n(D)$ in terms of their divided differences. A necessary and sufficient condition on $f\in K[X]$ to be integer-valued over $M_n(D)$ is that, for each $k$ less than $n$, the $k$-th divided difference of $f$ is integral-valued on every subset of the roots of any monic polynomial over $D$ of degree $n$. If in addition the intersection of the maximal ideals of finite index is $(0)$ then it is sufficient to check the above conditions on subsets of the roots of monic irreducible polynomials of degree $n$, that is, conjugate integral elements of degree $n$ over $D$.