# FWF - polynomials - Integer-valued polynomials

Project: Research project

# Project Details

### Description

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.
Status Finished 1/01/11 → 30/12/15