In computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating subset of an ideal I in a polynomial ring R. One can view it as a multivariate, non-linear generalization of:
The theory of Gröbner bases for polynomial rings was developed by Bruno Buchberger in 1965, who named them after his advisor Wolfgang Gröbner. The Association for Computing Machinery awarded him its 2007 Paris Kanellakis Theory and Practice Award for this work. An analogous concept for local rings was developed independently by Heisuke Hironaka in 1964, who named them standard bases. The analogous theory for free Lie algebras was developed by A. I. Shirshov in 1962 but his work remained largely unknown outside the Soviet Union.
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A Gröbner basis G of an ideal I in a polynomial ring R over a field is characterised by any one of the following properties, stated relative to some monomial order:
All these properties are equivalent; different authors use different definitions depending on the topic they choose. The last two properties that allow calculations in the factor ring R/I with the same facility as modular arithmetic. It is a significant fact of commutative algebra that Gröbner bases always exist, and can be effectively obtained for any ideal starting with a generating subset.
Multivariate division requires a monomial ordering, the basis depends on the monomial ordering chosen, and different orderings can give rise to radically different Gröbner bases. Two of the most commonly used orderings are lexicographic ordering, and degree reverse lexicographic order (also called graded reverse lexicographic order or simply total degree order). Lexicographic order eliminates variables, however the resulting Gröbner bases are often very large and expensive to compute. Degree reverse lexicographic order typically provides for the fastest Gröbner basis computations. In this order monomials are compared first by total degree, with ties broken by taking the smallest monomial with respect to lexicographic ordering with the variables reversed.
In most cases (polynomials in finitely many variables with complex coefficients or, more generally, coefficients over any field, for example), Gröbner bases exist for any monomial ordering. Buchberger's algorithm is the oldest and most well-known method for computing them. Other methods are the Faugère F4 algorithm, based on the same mathematics as the Buchberger algorithm, and involutive approaches, based on ideas from differential algebra. [1] There are also three algorithms for converting a Gröbner basis with respect to one monomial order to a Gröbner basis with respect to a different monomial order: the FGLM algorithm, the Hilbert Driven Algorithm and the Gröbner walk algorithm. These algorithms are often employed to compute (difficult) lexicographic Gröbner bases from (easier) total degree Gröbner bases.
A Gröbner basis is termed reduced if the leading coefficient of each element of the basis is 1 and no monomial in any element of the basis is in the ideal generated by the leading terms of the other elements of the basis. In the worst case, computation of a Gröbner basis may require time that is exponential or even doubly exponential in the number of solutions of the polynomial system (for degree reverse lexicographic order and lexicographic order, respectively). Despite these complexity bounds, both standard and reduced Gröbner bases are often computable in practice, and most computer algebra systems contain routines to do so.
The concept and algorithms of Gröbner bases have been generalized to modules over a polynomial ring, to free non-commutative polynomial rings and, by Weispfenning and his school, to solvable polynomial rings such as Weyl algebras.
Reduced Gröbner bases can be shown to be unique for any given ideal and monomial ordering, and are also often computable in practice. Thus one can determine if two ideals are equal by looking at their reduced Gröbner bases.
The reduction of a polynomial f by the multivariate division algorithm for an ideal using a Gröbner basis will yield 0 if and only if f is in the ideal. (By contrast, this is generally not true for a non-Gröbner basis with polynomials in more than one variable). This gives a test for determining whether or not a polynomial is in an ideal with a given set of generators.
If a Gröbner basis for an ideal I in
is computed relative to the lexicographic ordering with
the intersection of I with
is given by the intersection of the Gröbner basis with
In particular a polynomial f lies in
if and only if its leading term lies in this subring. This is known as the elimination property.
In particular, this gives us a method for solving simultaneous polynomial equations. If there are only finitely many solutions (over an algebraic closure of the field in which the coefficients lie) to the system of equations
we should be able to manipulate these equations to get something of the form
The elimination property says that if we compute a Gröbner basis for the ideal generated by {f1 – a1, ..., fm – am} relative to the right lexicographic ordering, then we can find the polynomial g as one of the elements of our basis. Furthermore, (taking k = n – 1) there will be another polynomial in the basis involving only xn – 1 and xn, so we can take our possible solutions for xn and find corresponding values for xn – 1. This lifting continues all the way up until we've found the values of all the variables.
The same elimination property can almost be used to convert parametric equations of polynomials into nonparametric equations. Given the equations
we compute a Gröbner basis for the ideal generated by
relative to any ordering that places polynomials involving t greater than those that don't: for example, lexicographic ordering with
Taking only the elements of the basis that do not involve the t variables, we get a set of equations describing not the original surface, but the smallest affine variety containing it.
and J is generated by some
then the intersection of I and J can be found by taking a Gröbner basis for the ideal generated by
relative to any lexicographic ordering that places t first, then taking only those terms not involving t. In particular, this allows us to calculate the least common multiple (and hence the greatest common divisor) of two polynomials f and g, since it is the generator of the intersection of the ideals generated by f and by g. This is true even if we do not know how to factor the polynomials! Also, note that for more than one variable the polynomial ring is not a Euclidean domain, so the Euclidean algorithm doesn't work here.