Buchberger's algorithm: Difference between revisions

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{{Short description|Algorithm for computing Gröbner bases}}

In the theory of [[multivariate polynomial]]s, '''Buchberger's algorithm''' is a method offor transforming a given set of [[Ideal (ring theory)#Ideal generated by a set|generators]] for a polynomial [[ring ideal|ideal]]polynomials into a [[Gröbner basis]], withwhich respectis toanother someset [[monomialof order]].polynomials Itthat washave inventedthe bysame Austriancommon [[mathematician]]zeros [[Brunoand Buchberger]].are Onemore canconvenient viewfor itextracting asinformation aon generalizationthese ofcommon thezeros. [[EuclideanIt algorithm]]was forintroduced univariateby [[GreatestBruno common divisor|GCDBuchberger]] computationsimultaneously andwith ofthe [[Gaussiandefinition elimination]]of forGröbner [[linearbases. system]]s.

[[Euclidean algorithm]] for polynomial [[Greatest common divisor]] computation and [[Gaussian elimination]] of [[system of linear equations|linear system]]s are special cases of Buchberger's algorithm when the number of variables or the degrees of the polynomials are respectively equal to one.

For other Gröbner basis algorithms, see ((slink|Gröbner basis#Algorithms and implementations}}.

== Algorithm ==