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Gröbner Bases: A Short Introduction for Systems Theorists
Bruno Buchberger
Research Institute for Symbolic Computation University of Linz, A4232 Schloss Hagenberg, Austria

[e-mail]−Linz.ac.at

Abstract. In this paper, we give a brief overview on Gröbner bases theory, addressed to novices without prior knowledge in the field. After explaining the general strategy for solving problems via the Gröbner approach, we develop the concept of Gröbner bases by studying uniquenss of polynomial division ("reduction"). For explicitly constructing Gröbner bases, the crucial notion of S polynomials is introduced, leading to the complete algorithmic solution of the construction problem. The algorithm is applied to examples from polynomial equation solving and algebraic relations. After a short discussion of complexity issues, we conclude the paper with some historical remarks and references.

1 Motivation for Systems Theorists
Originally, the method of Gröbner bases was introduced in [3, 4] for the algorithmic solution of some of the fundamental problems in commutative algebra (polynomial ideal theory, algebraic geometry). In 1985, on the invitation of N. K. Bose, I wrote a survey on the Gröbner bases method for his book on n dimensional systems theory, see [7]. Since then quite some applications of the Gröbner bases method have been found in systems theory. Soon, a special issue of the Journal of Multidimensional Systems and Signal Processing will appear that is entirely devoted to this topic, see [11]. Reviewing the recent literature on the subject, one detects that more and more problems in systems theory turn out to be solvable by the Gröbner bases method: factorization of multivariate polynomial matrices,
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solvability test and solution construction of unilateral and bilateral polynomial matrix equations, Bezout identity,
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design of FIR / IIR multidimensional filter banks,

stabilizability / detectability test and synthesis of feedback