A glimpse on numerical polynomial algebra,  or
what you have always wanted to know about Groebner$^*$ bases

Hans J. Stetter

Institute for Applied and Numerical Mathematics, Technical University of Vienna

stetter@aurora.anum.tuwien.ac.at

http://www.math.tuwien.ac.at/~stetter

Invited talk

Commutative Algebra is one of the few areas of computational mathematics whose numerical aspects have received little attention until recently. This contrasts with the importance of some of its fundamental tasks in Scientific Computation, e.g. the global solution of systems of multivariate polynomials. We will use an analysis of this task for an introduction into some aspects of numerical polynomials algebra which has become a highly active research area lately. In particular, we will demonstrate how matrix eigenproblems are at the heart of multivariate systems solving. On the way, we will see why Groebner basses have become a key tool of computational polynomial algebra; but we will also realize their problematic aspects in connection with numerical computation, when we have data of limited accuracy and use floating-point arithmetic. In conclusion, I will point out the fascination and challenge of the development of a numerical framework for polynomial algebra, where many concepts depend discontinuously on the data and where overdetermined representations are commonplace.

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$^*$  W. Gröbner, 1899-1980, Austrian mathematician, professor in Innsbruck

    From the homepage of Gerhard Wanner: ``Learned Mathematics from Wolfgang Gröbner''