Difference between revisions of "Gröbner basis"
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Gröbner bases are certain sets of multivariate polynomials with field coefficients. The importance of Gröbner bases stems from the fact that
i) many fundamental problems in algebraic geometry (commutative algebra, polynomial ideal theory) can be reduced by structurally easy algorithms to the construction of Gröbner bases; and
ii) there exists an algorithm by which for any given (finite) set of multivariate polynomials a Gröbner basis is constructed such that and generate the same polynomial ideal. (This implies, in particular, that, conceived as algebraic sets of equations, and have the same set of solutions.) Thus, Gröbner bases establish a uniform approach to the algorithmic solution of quite a few fundamental problems in algebraic geometry. New applications of Gröbner bases are discovered permanently. Also, the method of Gröbner bases has been carried over to domains other than polynomials over fields, e.g. polynomials with coefficients in certain rings, noncommutative polynomials, polynomial modules, and differential algebras. The algorithm that constructs Gröbner bases, various subalgorithms necessary for the construction of Gröbner bases, and various applications of Gröbner bases are routinely available in all modern mathematical software systems, as for example Mathematica and Maple.
A set of polynomials in , the polynomial ring over a field with indeterminates , is called a Gröbner basis if and only if all polynomials in the ideal generated by can be reduced to with respect to .
This definition involves the notion of "reduction" , which can be conceived as a generalized division for multivariate polynomials. For example, let
and
Then can be reduced, in one reduction step, with respect to by subtracting from , yielding a new polynomial :
Note that, by the reduction, the power product occurring in is replaced by "smaller" power products. In fact, a large class of total orderings on power products can be used in the Gröbnerbases approach for defining the notion of "smaller" . In the example, the polynomial can be reduced further with respect to by subtracting , yielding a new polynomial :
The polynomial cannot be reduced further with respect to and is called a reduced form of with respect to . A polynomial is reducible with respect to a set of polynomials if it is reducible with respect to some .
Gröbner bases can be characterized in many other ways, for example by saying that the above reduction relation with respect to has the Church–Rosser property, i.e. reduced forms are unique. The most important characterization of Gröbner bases is the content of the following main theorem: is a Gröbner basis if and only if, for all , the polynomial can be reduced to with respect to .
The polynomial of two polynomials and (without loss of generality, and are taken to have leading coefficient ) is defined as follows:
where
and
Here, denotes the leading power product of the polynomial , i.e. the power product in which is maximal with respect to the total ordering and denotes the least common multiple.
The above theorem is the crucial handle for constructing, for any given (finite) set of polynomials, a corresponding Gröbner basis , i.e. a Gröbner basis that generates the same ideal as : One first checks whether the condition formulated in the main theorem holds for . If this is the case, is already a Gröbner basis. If, however, the reduction of any of the polynomials yields a polynomial , then is added to the basis, new polynomials must be considered, etc., until the algorithm terminates. Termination of this algorithm can be proved by an invocation of either Hilbert's basis theorem (cf. Hilbert theorem) or Dickson's lemma. By the above main theorem, whose proof is combinatorial, it is easy to see that the resulting basis is a Gröbner basis corresponding to the initial set .
Important applications of Gröbner bases are: exact and complete solution of algebraic systems of equations, analysis of algebraic varieties, computation of Hilbert functions and Hilbert polynomials, complete solution of linear Diophantine systems of equations with polynomial coefficients, effective computation in polynomial residue class rings, structural analysis of hypercomplex systems, automated geometrical theorem proving, implicitization of parameter representations of manifolds, solution of various problems in invariant theory and integer programming, and solution of word problems in commutative semigroups.
The notion of a Gröbner basis, the main theorem, the algorithm based on the main theorem and some of the applications were introduced in [a2]. An easytoread summary of the theory is [a3]. The most comprehensive textbook on the subject is [a1].
References
[a1]  Th. Becker, V. Weispfenning, H. Kredel, "Gröbner bases: a computational approach to commutative algebra" , Springer (1993) 
[a2]  B. Buchberger, "An algorithmic criterion for the solvability of algebraic systems of equations" Aequationes Math. , 4 (1965) pp. 374–383 (This paper is a published version of the author's Ph.D. Thesis (Innsbruck, 1965) under the advice of Prof. W. Gröbner) 
[a3]  B. Buchberger, "Gröbner bases: an algorithmic method in polynomial ideal theory" N.K. Bose (ed.) , Recent Trends in Multidimensional Systems Theory , Reidel (1985) pp. 184–232 
Gröbner basis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gr%C3%B6bner_basis&oldid=22529