The ring of integer-valued polynomials was studied at the beginning of XX century by Polya. We consider polynomials with rational coefficients whose image over the ring of integers is contained in the integers, that is

$${\rm Int}(\mathbb{Z})=\{f\in\mathbb Q[X] \mid f(\mathbb Z)\subset \mathbb Z\}.$$

Obviously, the ring of polynomials with integer coefficients is contained in ${\rm Int}(\mathbb{Z})$ and this containment is strict: for example consider the polynomial $X(X-1)/2$. Using a modern terminology, Polya proved that this ring has a free basis as a $\mathbb{Z}$-module made up by the so-called binomial polynomials: $f_n(X)=X(X-1)\ldots(X-n+1)/n!$. This means that, given an integer-valued polynomial $f(X)$ of degree $n$, there exist integer coefficients $c_i$, for $i=0,...,n$ such that $f(X)=c_0f_0(X)+...+c_nf_n(X)$ and these coefficients $c_i$ are uniquely determined by $f(X)$. We notice that for each $n\in\mathbb{N}$, the polynomial $f_n(X)$ has degree $n$. Such a set of polynomials $\{f_n(X)\}_{n\in\mathbb{N}}$ in ${\rm Int}(\mathbb{Z})$ is usually called a regular basis for the ring ${\rm Int}(\mathbb{Z})$.

More in general, given a domain $D$ with quotient field $K$, we denote by ${\rm Int}(D)$ the ring of polynomials with coefficients in $K$ such that the image of $f(X)$ over $D$ is contained in $D$, that is

$${\rm Int}(D)=\{f\in K[X] \mid f(D)\subset D\}$$

(in order to avoid trivialities, we will always assume that $D$ is not a field). The book of Cahen-Chabert, „Integer-valued Polynomials“ is an exhaustive resource on the argument.

During the last years the interest towards ring of integer-valued polynomials has grown. Many other aspects have been studied or introduced as new. Here is a list of the conferences related to these topics, where the researchers of this project have participated, bringing their own contributions:

From the conference held in Graz, a special volume for the conference proceedings will be published by Springer. For informations, see here.

Last edited: in 2012 by Giulio Peruginelli