Universität Innsbruck
Institut für Mathematik

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	This is the homepage of the FWF-Project P16641

Relation Modules for Conjugate Algebraic Numbers

at the Institute of Mathematics of the University of Innsbruck.

Abstract

The aim of the Project is to develop a better understanding of the mathematical structures associated with the algebraic relations between the zeros of a univariate, separable polynomial  f  with coefficients in an arbitrary field K. Clearly the most important case emerges if we choose  f  to be irreducible and K to be the rational numbers. If x is the vector of all zeros of  f  in the algebraic closure of K, then a polynomial vanishing at x is called a relation. All relations form the relation ideal of  f. It is in the center of our considerations. Of special interest are also the so-called linear relations. These are relations of degree one, i.e. they express K-linear dependence of the zeros of  f .  For the investigation of these special relations representation theory proved to be a very powerful tool. Other methods and theories used are Galois theory , algebraic geometry, Gr�bner bases and invariant theory .

History

The project started in October 2003. Besides the non-stop  engagement of the two associate Professors Kurt Girstmair and Franz Pauer, Mathias Lederer with his diploma thesis and his PhD thesis, made significant contributions in the beginning. He now holds an assistant position at the University of Bielefeld. For his present activities see here. In December 2004 Michael Wibmer got involved in the project. He has finished his diploma thesis and is working on PhD level now.

Selected publications

Michael Wibmer, Gr�bner Bases for Families of Affine or Projective Schemes, preprint, 2006, pdf

Mathias Lederer, The vanishing ideal of a finite set of closed points in affine space, preprint, 2006, pdf

Kurt Girstmair, The Galois Relation x1=x2+x3 and Fermat over Finite Fields, to appear in Acta Arithmetica, 2006, pdf

Larry Smith, Michael Wibmer, On the Dimension of Coinvariants of Permutation Representations, to appear in Monatshefte für Mathematik, 2006, pdf

Mathias Lederer, A determinant-like formula for the Kostka numbers, to appear in Annals of Combinatorics, 2005, pdf

Kurt Girstmair, Franz Pauer, Michael Wibmer, On invariant relations between zeros of polynomials, Communications in Algebra, Volume 33, Number 7, 2157-2166, 2005

Michael Wibmer , Invariante Relationen zwischen konjugierten algebraischen Zahlen, Diplomarbeit, 2005, pdf

Mathias Lederer, Relation ideals and the Buchberger-M�ller Algorithm, preprint, 2005, pdf

Mathias Lederer, Explicit constructions in splitting fields of polynomials, Riv. Mat. Univ. Parma (7) 3*, 233-244, 2004, pdf

Mathias Lederer,  Relationenmoduln f�r konjugierte algebraische Zahlen, Dissertation, 2004, pdf

Mathias Lederer, Explizite Konstruktionen in Zerf�llungsk�rpern von Polynomen, Diplomarbeit, 2002, pdf

For more publications see the personal homepages of the project members: Kurt Girstmair, Franz Pauer, Mathias Lederer. A short description of the project on the FWF-server can be found here.

last modified on 5/11/2007 15:20:32 by Michael Wibmer