## Website in connection with FWF Project „Integer-valued polynomials“ (grant P 23245-N18).

Background

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:

• 19-21 May 2010: Commutative Ring Theory Days 2010, Universita' di Roma Tre, Italy.

• 29 November-3 December 2010: Troisieme Rencontre sur les Polynomes a valeurs entieres, Cirm, Marseille, France. Website.

• 4-8 June 2012: Commutative Rings and their Modules, 2012, Bressanonone, Italy, Website.

• 16-22 December, 2012, Commutative rings, integer-valued polynomials and polynomial functions in Graz, Austria (Mini-Courses+Conference; organized by the same research group of this project). (Website: Mini-Courses (Dec 16-18), Conference (Dec. 19-22)).

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

Recent developments: integer-valued polynomials on algebras

Many authors (Frisch, Werner, Loper, Tartarone, Cigliola) have introduced and studied the new class of ring of integer-valued polynomials over an algebra, starting a totally new stream of studies. In order to define such rings, suppose that $A$ is a $D$-algebra which is torsion-free. Such a hypothesis allows us to embed $D$ into $A$ and we will suppose that $D$ is contained in $A$. If we consider the tensor product $B=A\otimes_D K$, we can in the same way embed $A$ and $K$ into $B$ via the canonical mapping. Given a polynomial $f(X)$ in $K[X]$ we can then evaluate $f(X)$ over the elements of $B$. The set of $f\in K[X]$ such that $f(a)$ is contained in $A$ for each $a\in A$ is denoted by ${\rm Int}_D(A)$. We can easily see that this set comprises a ring. Moreover, if we also suppose that $K\cap A=D$, we obtain that such a ring is contained in the usual ring of integer-valued polynomials over $D$.

Examples of such rings are the following:

1. Ring of integer-valued polynomials on the algebra $M_n(D)$ of matrices over $D$, denoted by ${\rm Int}_K(M_n(D))$.

2. Ring of integer-valued polynomials over quaternions.

3. Ring of polynomials with rational coefficients which are integer-valued over a ring of integers of a given number field.

In the first two examples the $D$-algebra $A$ is non-commutative. The ring of the last example, denoted by ${\rm Int}_{\mathbb{Q}}(O_K)$ is precisely the contraction to $\mathbb{Q}[X]$ of the subring of $K[X]$ of integer-valued polynomials over $O_K$, that is ${\rm Int}(O_K)$. Such a ring is a first example of polynomials with coefficient in a field $K$ which are integer-valued over a domain not contained in $K$ itself.

Recently, Evrard, Fares and Johnson considered the sub-algebra of the triangular matrices and they have studied the ring of integer-valued polynomials over that sub-algebra (\cite{2}). In particular they prove that this ring coincides with the ring of polynomials whose divided differences are integer-valued on every vector of elements of $D$.

Loper and Werner studied another kind of ring of integer-valued polynomials. Given a positive integer n, let $\mathcal{A}_n$ be the set of all algebraic integers over $\mathbb{Z}$ of degree bounded by n (inside a fixed algebraic closure of the rationals). We denote by

$${\rm Int}_{\mathbb{Q}}(\mathcal{A}_n)=\{f\in\mathbb{Q}[X] \mid f(\mathcal{A}_n)\subset\mathcal{A}_n\}$$

which is called the ring of polynomials with rational coefficients which are integral-valued over $\mathcal{A}_n$. Obviously, such a ring is strictly contained in ${\rm Int}(\mathbb{Z})$. They prove that such a ring is a Prüfer domain (so in particular $\mathbb{Z}[X]$ is strictly contained in it) and it is equal to the integral closure of the ring ${\rm Int}(M_n(\mathbb{Z}))$. The ring ${\rm Int}_{\mathbb{Q}}(\mathcal{A}_n)$ can be represented in this way:

$${\rm Int}_{\mathbb{Q}}(\mathcal{A}_n)=\bigcap_{[K:\mathbb{Q}]\leq n}{\rm Int}_{\mathbb{Q}}(O_K)$$

where the intersection is taken over the family of number fields $K$ of degree bounded by $n$. This means that a polynomial in ${\rm Int}_{\mathbb{Q}}(\mathcal{A}_n)$ is integral-valued over the ring of integers of every such number field.

Results obtained

1.G. Peruginelli studied the ideal of polynomials in $\mathbb{Z}[X]$ having fixed divisor divisible by a prime power („Primary decomposition of the ideal of polynomials whose fixed divisor is divisible by a prime power, accepted by Journal of Algebra). Given a polynomial $f\in\mathbb{Z}[X]$, we study here the image set of $f(X)$ over the integers by considering the ideal of $\mathbb{Z}$ generated by the values of $f(X)$ over the integers. This ideal is usually called the fixed divisor of $f(X)$. Given a positive integer $m$, we consider the ideal $I_m$ of $\mathbb{Z}[X]$ made up by those polynomial whose fixed divisor is divisible by $m$. We easily show that we can reduce the study of this ideal to the case of $m$ equal to a prime power $p^n$. We describe here explicitly the ideal $I_{p^n}$ by means of its primary decomposition, given a set of generators of each of its primary components, which are permuted by $\mathbb{Z}[X]$-automorphisms. The set of these primary components turns out to be equal to the set of polynomial ideals which are congruent to zero modulo $p^n$ on the residue classes modulo $p$. Further application of this result are in the study of the factorization in ${\rm Int}(\mathbb{Z})$.

2. G. Peruginelli generalized the result of Evrard, Fares and Johnson about integer-valued polynomials on triangular matrices $T_n(D)$ to the ring of integer-valued polynomials over the full algebra of matrices $M_n(D)$, where $D$ is an integrally closed domain („Integer-valued polynomials over matrices and divided differences“, published in Monatshefte für Mathematik). We give sufficient and necessary conditions on the divided differences of a polynomial $f\in K[X]$ so that $f(X)$ is integer-valued over $M_n(D)$. The conditions are the following: for each $0\leq k< n$, the $k$-th divided difference of $f(X)$ (which is a polynomial with coefficients in $K$ in $k+1$ variables) is integer-valued on every subset of $k+1$ roots of every monic polynomial $p(X)$ of degree $n$ in $D[X]$, where the roots are considered in a fixed algebraic closure of $K$. This means that the divided differences of $f(X)$ are integer-valued over vectors of integral elements over $D$ which do not belong to $K$.

3. Starting from the results obtained by Loper and Werner, G. Peruginelli proved that if a polynomial $f\in\mathbb{Q}[X]$ is integral-valued over the set A_n of the algebraic integers over $\mathbb{Z}$ of degree equal to a fixed positive integer $n$, then $f(X)$ is integral-valued over every algebraic integer of degree strictly smaller (see „ Integral-valued polynomials over sets of algebraic integers of bounded degree“, submitted). Using a well-known terminology we say that A_n is polynomially dense in $\mathcal{A}_n$. We can thus write that ${\rm Int}_{\mathbb{Q}}(\mathcal{A}_n)={\rm Int}_{\mathbb{Q}}(A_n,\mathcal{A}_n)$, where

$${\rm Int}_{\mathbb{Q}}(A_n,\mathcal{A}_n)=\{f\in\mathbb{Q}[X] | f(A_n)\subset\mathcal{A}_n\}$$

is the ring of integer-valued polynomials over $A_n$.

Conference „Commutative ring theory, integer-valued polynomials and polynomial functions“

December 16-22, 2012, Graz, Austria.

Dec. 16-18: Mini-Courses

Dec. 19-22: Conference

Researchers supported by FWF grant P 23245-N18:

Contact: commring@TUGraz.at