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 $A_n$.

Last edited: in 2012 by Giulio Peruginelli