Gröbner Bases: A short introduction for systems theorists

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. © 2011 Springer-Verlag Berlin Heidelberg.