Integer-valued polynomials

In this project, three mathematicians at TU Graz - Sophie Frisch, Giulio Peruginelli, and doctoral student Roswitha Rissner, propose to do research on integer-valued polynomials and at the same time refine the number-theoretic and ring-theortic methods that are needed for this research. An integer-valued polynomial is a polynomial (in one or several variables) with rational coefficients that takes an integer value whenever integers are substituted for the variables. More generally, one considers polynomials with coefficients in the quotient field K of an integral domain D that take values in D whenever elements of D are substituted for the variables. Rings of integer-valued polynomials have interesting properties both from a ring-theoretic and from a number-theoretic point of view. For well-behaved D, including rings of algebraic integers in number fields, the ring of integer-valued polynomials is a Prüfer ring, and one can interpolate arbitrary functions from D^n to D by integer-valued polynomials. Also, rings of integer-valued polynomials enjoy a natural generalization of Hilbert`s Nullstellensatz. The current project aims firstly to explore the potential of integer-valued polynomials for parametrization of integer solutions of Diophantine equations. Two of the proponents have already obtained results in this directions, so, for instance, that the set of integer Pythagoraean triples can be parametrized by a single triple of integer-valued polynomials, while at least two triples are needed for a parametrization by polynomials with integer coefficients. Secondly, the project is about investigating integer-valued polynomials on algebras, for instance, the ring of polynomials with rational coefficients (in one variable) that map every integer n by n matrix to an integer matrix.

In the classical sense, integer-valued polynomials are polynomials with rational coefficients that map all integers to integers. Already in the 17th century, Newton used this kind of polynomials to interpolate functions. At the beginning of the 20th century, Plya, Ostrowski and Skolem discovered their role in number theory.In addition to their usefulness for interpolation, integer-valued polynomials have other interesting algebraic and number theoretic properties. For example, they satisfy a p-adic version of the Stone-Weierstrass property which make them suitable for approximation purposes, and analogues of Hilberts Nullstellensatz, the so-called Skolem properties.We researchers of this project, Sophie Frisch, Giulio Peruginelli and Roswitha Rissner, worked in the more general context of integer-valued polynomials on arbitrary domains. We narrowed down the gap between sufficient and necessary conditions for Skolem properties to almost nothing, the first significant advance on this question in decades.Also, we were among the first to investigate polynomials which are integervalued on matrix rings or more general algebras. We showed that integervalued polynomials with coefficients in a non-commutative matrix algebra can be expressed as matrices whose entries are polynomials with coefficients in a commutative ring. In the special case of upper triangular matrix algebras, integer-valued polynomials coincide with polynomials whose divided differences up to order n are integer-valued. These polynomials have applications, discovered by Bhargava, to Banach spaces of n-times continuously differentiable functions on a compact subset of a local field. We also showed various other number theoretic and ring theoretic results, well-received by the international scientific community, concerning integer-valued polynomials, their factorization, their overrings and the Prüfer property.