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 Gröbner basis (Definition)
 Definition of monomial orderings and support: Let be a field, and let be the set of monomials in , the polynomial ring in indeterminates. A monomial ordering is a total ordering on which satisfies implies that for all . for all . In practice, for any , we associate to the string and compare monomials by looking at orderings on these -tuples. Example. An extremely prevalent example of a monomial ordering is given by the standard lexicographical ordering of strings. Other examples include graded lexicographic ordering and graded reverse lexicographic ordering. Henceforth, assume that we have fixed a monomial ordering. Define the support of , denoted , to be the set of terms with . Then define . A partial order on : We can extend our monomial ordering to a partial ordering on as follows: Let . If , we say that if . It can be shown that: The relation defined above is indeed a partial order on Every descending chain with is finite. A division algorithm for : We can then formulate a division algorithm for : Let be an ordered -tuple of polynomials, with . Then for each , there exist with unique, such that For each , does not divide any monomial in . Furthermore, if for some , then . Definition of Gröbner basis: Let be a nonzero ideal of . A finite set of polynomials is a Gröbner basis for if for all with there exists such that . Existence of Gröbner bases: Every ideal other than the zero ideal has a Gröbner basis. Additionally, any Gröbner basis for is also a basis of .

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 Also defines: monomial ordering, Gröbner basis

Cross-references: basis, zero ideal, bases, ideal, divide, polynomials, division algorithm, finite, chain, relation, terms, fixed, orderings, string, associate, implies, satisfies, total ordering, indeterminates, polynomial ring, monomials, field, support
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This is version 25 of Gröbner basis, born on 2002-09-23, modified 2006-10-04.
Object id is 3470, canonical name is GrobnerBasis.
Accessed 4526 times total.

Classification:
 AMS MSC: 13P10 (Commutative rings and algebras :: Computational aspects of commutative algebra :: Polynomial ideals, Gröbner bases)