# 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.

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 |
---|---|

Effective start/end date | 1/01/11 → 30/12/15 |