FWF - polynomials - 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.