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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.