Gröbner Bases: A Short Introduction for Systems Theorists

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, solvability test and solution construction of unilateral and bilateral polynomial matrix equations, Bezout identity, design of FIR / IIR multidimensional filter banks, stabilizability / detectability test and synthesis of feedback stabilizing compensator / asymptotic observer, synthesis of deadbeat or asymptotic tracking controller / regulator, constructive solution to the nD polynomial matrix completion problem, computation of minimal left annhilators / minimal right annhilators, elimination of variables for latent variable representation of a behaviour, computation of controllable part; controllability test, observability test, computation of transfer matrix and "minimal realization", solution of the Cauchy problem for discrete systems, testing for inclusion; addition of behaviors, test zero / weak zero / minor primeness, finite dimensionality test, computation of sets of poles and zeros; polar decomposition, achievability by regular interconnection, computation of structure indices. In [11], I gave the references to these applications and I also presented an easy introduction to the theory of Gröbner bases by giving a couple of worked out examples. In this paper, I will give an introduction to Gröbner bases in the style of a flyer for promotion that just answers a couple of immediate questions on the theory for newcomers. Thus, [11] and the present paper are complementary and, together, they may provide a quick and easy introduction to Gröbner bases theory, while [7] provides a quick guide to the application of the method to fundamental problems in commutative algebra. 2 Why is Gröbner Bases Theory Attractive? Gröbner bases theory is attractive because the main problem solved by the theory can be explained in five minutes (if one knows the operations of addition and multiplication on polynomials), the algorithm that solves the problem can be learned in fifteen minutes (if one knows the operations of addition and multiplication on polynomials), the theorem on which the algorithm is based is nontrivial to (invent and to) prove, many problems in seemingly quite different areas of mathematics can be reduced to the problem of computing Gröbner bases. 3 What is the Purpose of Gröbner Bases Theory? The method (theory plus algorithms) of Gröbner bases provides a uniform approach to solving a wide range of problems expressed in terms of sets of multivariate polynomials. Areas in which the method of Gröbner bases has bee applied successfully are: algebraic geometry, commutative algebra, polynomial ideal theory, invariant theory, automated geometrical theorem proving, coding theory, integer programming, partial differential equations, hypergeometric functions, symbolic summation, statistics, non commutative algebra, numerics (e.g. wavelets construction), and systems theory. The book [9] includes surveys on the application of the Gröbner bases method for most of the above areas. In commutative algebra, the list of problems that can be attacked by the Gröbner bases approach includes the following: solvability and solving of algebraic systems of equations, ideal and radical membership decision, effective computation in residue class rings modulo polynomial ideals, linear diophantine equations with polynomial coefficients ("syzygies"), Hilbert functions, algebraic relations among polynomials, implicitization, inverse polynomial mappings. 4 How Can Gröbner Bases Theory be Applied? The general strategy of the Gröbner bases approach is as follows: Given a set F of polynomials in K x1 , , xn (that describes the problem at hand) we transform F into another set G of polynomials "with certain nice properties" (called a "Gröbner basis") such that F and G are "equivalent" (i.e. generate the same ideal). From the theory of GB we know: Because of the "nice properties of Gröbner bases", many problems that are difficult for general F are "easy" for Gröbner bases G. There is an algorithm for transforming an arbitrary F into an equivalent Gröbner basis G . The solution of the problem for G can often be easily translated back into a solution of the problem for F. Hence, by the properties of Gröbner bases and the possibility of transforming arbitrary finite polynomial sets into Gröbner bases, a whole range of problems definable in terms of finite polynomial sets becomes algorithmically solvable. 5 What are Gröbner Bases? 5.1 Division ("Reduction") of Multivariate Polynomials We first need the notion of division (or "reduction") for multivariate polynomials. Consider, for example, the following bivariate polynomials g , f1 , and f2 , and the following polynomial set F : (1) g x y 3 x y 5 x, (2) f1 x y 2 y, f2 2 y x , (3) F f1 , f2 . The monomials in these polygonomials are ordered. There are infinitely many orderings that are "admissible" for Gröbner bases theory. The most important ones are the lexicographic orderings and the orderings that, first, order power products by their degree and, then, lexicographically. In the example above, the monomials are ordered lexicographically with y ranking higher than x and are presented in descending order from left to right. The highest (left most) monomial in a polynomial is called the "leading" monomial in the polynomial. The monomials in these polygonomials are ordered. There are infinitely many orderings that are "admissible" for Gröbner bases theory. The most important ones are the lexicographic orderings and the orderings that, first, order power products by their degree and, then, lexicographically. In the example above, the monomials are ordered lexicographically with y ranking higher than x and are presented in descending order from left to right. The highest (left most) monomial in a polynomial is called the "leading" monomial in the polynomial. One possible division ("reduction") step that "reduces the polymomial g modulo f1 " proceeds as follows: (4) h g 3 y f1 5 x 6 y x y , i.e. in a reduction step of g modulo f1 , by subtracting a suitable monomial multiple of f1 from g, one of the monomials of g should cancel against the leading monomial of 3 y f1 . We write (5) g f1 h for this situation (read: "g reduces to h modulo f1 "). 5.2 In General, Many Reductions are Possible Given a set F of polynomials and a polynomial g, many different reductions of g modulo polynomials in F may be possible. For example, for g and F as above, we also have (6) h2 g x y f1 5 x 3 x y 2 x y , (7) h3 g 1 2 x y f2 5 x x4 y 2 3 x y , and, hence, (8) g f1 h2 , (9) g f2 h3 . 5.3 Multivariate Polynomial Division Always Terminates But is Not Unique We write (10) g F h if (11) g f h for some f F , and we write (12) g F h if g reduces to h by finitely many reduction steps w.r.t. F . Also, we write