Gröbner basis


In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field. A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps.
Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common divisors, and
Gaussian elimination for linear systems.
Gröbner bases were introduced by Bruno Buchberger in his 1965 Ph.D. thesis, which also included an algorithm to compute them. He named them after his advisor Wolfgang Gröbner. In 2007, Buchberger received the Association for Computing Machinery's Paris Kanellakis Theory and Practice Award for this work.
However, the Russian mathematician Nikolai Günther had introduced a similar notion in 1913, published in various Russian mathematical journals. These papers were largely ignored by the mathematical community until their rediscovery in 1987 by Bodo Renschuch et al. An analogous concept for multivariate power series was developed independently by Heisuke Hironaka in 1964, who named them standard bases. This term has been used by some authors to also denote Gröbner bases.
The theory of Gröbner bases has been extended by many authors in various directions. It has been generalized to other structures such as polynomials over principal ideal rings or polynomial rings, and also some classes of non-commutative rings and algebras, like Ore algebras.

Tools

Polynomial ring

Gröbner bases are primarily defined for ideals in a polynomial ring over a field. Although the theory works for any field, most Gröbner basis computations are done either when is the field of rationals or the integers modulo a prime number.
In the context of Gröbner bases, a nonzero polynomial in is commonly represented as a sum where the are nonzero elements of, called coefficients, and the are monomials of the form where the are nonnegative integers. The vector is called the exponent vector of the monomial. When the list of the variables is fixed, the notation of monomials is often abbreviated as
Monomials are uniquely defined by their exponent vectors, and, when a monomial ordering is fixed, a polynomial is uniquely represented by the ordered list of the ordered pairs formed by an exponent vector and the corresponding coefficient. This representation of polynomials is especially efficient for Gröbner basis computation in computers, although it is less convenient for other computations such as polynomial factorization and polynomial greatest common divisor.
If is a finite set of polynomials in the polynomial ring, the ideal generated by is the set of linear combinations of elements of with coefficients in ; that is the set of polynomials that can be written with

Monomial ordering

All operations related to Gröbner bases require the choice of a total order on the monomials, with the following properties of compatibility with multiplication. For all monomials,,,
  2. .
A total order satisfying these condition is sometimes called an admissible ordering.
These conditions imply that the order is a well-order, that is, every strictly decreasing sequence of monomials is finite.
Although Gröbner basis theory does not depend on a particular choice of an admissible monomial ordering, three monomial orderings are especially important for the applications:
Gröbner basis theory was initially introduced for the lexicographical ordering. It was soon realised that the Gröbner basis for degrevlex is almost always much easier to compute, and that it is almost always easier to compute a lex Gröbner basis by first computing the degrevlex basis and then using a "change of ordering algorithm". When [|elimination] is needed, degrevlex is not convenient; both lex and lexdeg may be used but, again, many computations are relatively easy with lexdeg and almost impossible with lex.

Basic operations

Leading term, coefficient and monomial

Once a monomial ordering is fixed, the terms of a polynomial are naturally ordered by decreasing monomials. This makes the representation of a polynomial as a sorted list of pairs coefficient–exponent vector a canonical representation of the polynomials.
The first term of a polynomial for this ordering and the corresponding monomial and coefficient are respectively called the leading term, leading monomial and leading coefficient and denoted, in this article, and.
Most polynomial operations related to Gröbner bases involve the leading terms. So, the representation of polynomials as sorted lists make these operations particularly efficient.

Polynomial operations

The other polynomial operations involved in Gröbner basis computations are also compatible with the monomial ordering; that is, they can be performed without reordering the result:
  • The addition of two polynomials consists in a merge of the two corresponding lists of terms, with a special treatment in the case of a conflict.
  • The multiplication of a polynomial by a scalar consists of multiplying each coefficient by this scalar, without any other change in the representation.
  • The multiplication of a polynomial by a monomial consists of multiplying each monomial of the polynomial by. This does not change the term ordering by definition of a monomial ordering.

    Divisibility of monomials

Let and be two monomials, with exponent vectors and
One says that divides, or that is a multiple of, if for every ; that is, if is componentwise not greater than. In this case, the quotient is defined as In other words, the exponent vector of is the componentwise subtraction of the exponent vectors of and.
The greatest common divisor of and is the monomial whose exponent vector is the componentwise minimum of and. The least common multiple is defined similarly with instead of.
One has

Reduction

The reduction of a polynomial by other polynomials with respect to a monomial ordering is central to Gröbner basis theory. It is a generalization of both row reduction occurring in Gaussian elimination and division steps of the
Euclidean division of univariate polynomials. When completed as much as possible, it is sometimes called multivariate division although its result is not uniquely defined.
Lead-reduction is a special case of reduction that is easier to compute. It is fundamental for Gröbner basis computation, since general reduction is needed only at the end of a Gröbner basis computation, for getting a reduced Gröbner basis from a non-reduced one.
Let an admissible monomial ordering be fixed, to which refers every monomial comparison that will occur in this section.
A polynomial is lead-reducible by another polynomial if the leading monomial is a multiple of. The polynomial is reducible by if some monomial of is a multiple of.
Suppose that is reducible by, and let be a term of such that the monomial is a multiple of. A one-step reduction of by consists of replacing by
This operation removes the monomial from without changing the terms with a monomial greater than . In particular, a one step lead-reduction of produces a polynomial all of whose monomials are smaller than.
Given a finite set of polynomials, one says that is reducible or lead-reducible by if it is reducible or lead-reducible, respectively, by at least one element of. In this case, a one-step reduction of by is any one-step reduction of by an element of.
The reduction of by consists of iterating one-step reductions until getting a polynomial that is irreducible by. It is sometimes called a normal form of by. In general this form is not uniquely defined because there are, in general, several elements of that can be used for reducing ; this non-uniqueness is the starting point of Gröbner basis theory.
The definition of the reduction shows immediately that, if is a normal form of by, one has
where is irreducible by and the are polynomials such that In the case of univariate polynomials, if consists of a single element, then is the remainder of the Euclidean division of by, and is the quotient. Moreover, the division algorithm is exactly the process of lead-reduction. For this reason, some authors use the term multivariate division instead of reduction.

Non uniqueness of reduction

In the example that follows, there are exactly two complete lead-reductions that produce two very different results. The fact that the results are irreducible is specific to the example, although this is rather common with such small examples.
In this two variable example, the monomial ordering that is used is the lexicographic order with and we consider the reduction of, by with
For the first reduction step, either the first or the second term of may be reduced. However, the reduction of a term amounts to removing this term at the cost of adding new lower terms; if it is not the first reducible term that is reduced, it may occur that a further reduction adds a similar term, which must be reduced again. It is therefore always better to reduce first the largest reducible term; that is, in particular, to lead-reduce first until getting a lead-irreducible polynomial.
The leading term of is reducible by and not by So the first reduction step consists of multiplying by and adding the result to :
The leading term of is a multiple of the leading monomials of both and So, one has two choices for the second reduction step. If one chooses one gets a polynomial that can be reduced again by
No further reduction is possible, so is a complete reduction of.
One gets a different result with the other choice for the second step:
Again, the result is irreducible, although only lead reductions were done.
In summary, the complete reduction of can result in either or
It is for dealing with the problems set by this non-uniqueness that Buchberger introduced Gröbner bases and -polynomials. Intuitively, may be reduced to This implies that belongs to the ideal generated by. So, this ideal is not changed by adding to, and this allows more reductions. In particular, can be reduced to by and this restores the uniqueness of the reduced form.
Here Buchberger's algorithm for Gröbner bases would begin by adding to the polynomial
This polynomial, called -polynomial by Buchberger, is the difference of the one-step reductions of the least common multiple of the leading monomials of and, by and respectively:
In this example, one has This does not complete Buchberger's algorithm, as gives different results, when reduced by or