Gröbner bases and systems theory. (English) Zbl 1088.93500

Buchberger, Bruno

Multidimensional Syst. Signal Process. 12, No. 3-4, 223-251 (2001).

Summary: We present the basic concepts and results of Gröbner bases theory for readers working or interested in systems theory. The concepts and methods of Gröbner bases theory are presented by examples. No prerequisites, except some notions of elementary mathematics, are necessary for reading this paper. The two main properties of Gröbner bases, the elimination property and the linear independence property, are explained. Most of the many applications of Gröbner bases theory, in particular applications in systems theory, hinge on these two properties. Also, an algorithm based on Gröbner bases for computing complete systems of solutions (“syzygies”) for linear diophantine equations with multivariate polynomial coefficients is described. Many fundamental problems of systems theory can be reduced to the problem of syzygies computation.

MSC:

93A05	Axiomatic systems theory
68W30	Symbolic computation and algebraic computation
93-02	Research exposition (monographs, survey articles) pertaining to systems and control theory
93B25	Algebraic methods

Keywords:

Gröbner bases; algorithmic systems theory; computer algebra; algebraic algorithms; polynomial ideals; elimination; residue class rings; syzygies; polynomial diophantine equations

Software:

Theorema