# Difference between revisions of "ApCoCoA-1:NCo.IsGB"

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<title>NCo.IsGB</title> | <title>NCo.IsGB</title> | ||

<short_description> | <short_description> | ||

− | Check whether a finite | + | Check whether a finite LIST of non-zero polynomials in a free monoid ring is a Groebner basis. |

<par/> | <par/> | ||

− | Note that, given an ideal <tt>I</tt> and | + | Note that, given an ideal <tt>I</tt> and a word ordering <tt>Ordering</tt>, a set of non-zero polynomials <tt>G</tt> is called a <em>Groebner basis</em> of <tt>I</tt> with respect to <tt>Ordering</tt> if the leading word set <tt>LW{G}</tt> generates the leading word ideal <tt>LW(I)</tt>. The function checks whether a given finite LIST of non-zero polynomials <tt>G</tt> is a Groebner basis by using the <tt>Buchberger Criterion</tt>, i.e. <tt>G</tt> is a Groebner basis if the S-polynomials of all obstructions have the zero normal remainder with respect to <tt>G</tt>. |

</short_description> | </short_description> | ||

<syntax> | <syntax> | ||

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<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | <em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | ||

<par/> | <par/> | ||

− | Please set ring environment <em>coefficient field</em> <tt>K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>NCo.SetFp</ref>, <ref>NCo.SetX</ref> and <ref>NCo.SetOrdering</ref>, respectively, before | + | Please set ring environment <em>coefficient field</em> <tt> K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>NCo.SetFp</ref>, <ref>NCo.SetX</ref> and <ref>NCo.SetOrdering</ref>, respectively, before using this function. The default coefficient field is <tt>Q</tt>, and the default ordering is the length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions. |

<itemize> | <itemize> | ||

− | <item>@param <em>G</em>: a LIST of non-zero polynomials in <tt>K<X></tt>. Each polynomial is represented as a LIST of monomials, which are | + | <item>@param <em>G</em>: a LIST of non-zero polynomials in <tt>K<X></tt>. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <tt><X></tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]].</item> |

− | <item>@return: a BOOL | + | <item>@return: a BOOL, which is True if <tt>G</tt> is a Groebner basis with respect to the current word ordering and False otherwise.</item> |

</itemize> | </itemize> | ||

<example> | <example> | ||

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<types> | <types> | ||

<type>apcocoaserver</type> | <type>apcocoaserver</type> | ||

+ | <type>ideal</type> | ||

<type>groebner</type> | <type>groebner</type> | ||

− | |||

<type>non_commutative</type> | <type>non_commutative</type> | ||

</types> | </types> |

## Revision as of 17:01, 29 April 2013

## NCo.IsGB

Check whether a finite LIST of non-zero polynomials in a free monoid ring is a Groebner basis.

Note that, given an ideal `I` and a word ordering `Ordering`, a set of non-zero polynomials `G` is called a *Groebner basis* of `I` with respect to `Ordering` if the leading word set `LW{G}` generates the leading word ideal `LW(I)`. The function checks whether a given finite LIST of non-zero polynomials `G` is a Groebner basis by using the `Buchberger Criterion`, i.e. `G` is a Groebner basis if the S-polynomials of all obstructions have the zero normal remainder with respect to `G`.

### Syntax

NCo.IsGB(G:LIST):BOOL

### Description

*Please note:* The function(s) explained on this page is/are using the *ApCoCoAServer*. You will have to start the ApCoCoAServer in order to use it/them.

Please set ring environment *coefficient field* ` K`, *alphabet* (or set of indeterminates) `X` and *ordering* via the functions NCo.SetFp, NCo.SetX and NCo.SetOrdering, respectively, before using this function. The default coefficient field is `Q`, and the default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

@param

*G*: a LIST of non-zero polynomials in`K<X>`. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in`<X>`and C is the coefficient of W. For example, the polynomial`f=xy-y+1`is represented as F:=[[1,"xy"], [-1, "y"], [1,""]].@return: a BOOL, which is True if

`G`is a Groebner basis with respect to the current word ordering and False otherwise.

#### Example

NCo.SetX(<quotes>xyt</quotes>); F1 := [[1,<quotes>xx</quotes>], [-1,<quotes>yx</quotes>]]; F2 := [[1,<quotes>xy</quotes>], [-1,<quotes>ty</quotes>]]; F3 := [[1,<quotes>xt</quotes>], [-1, <quotes>tx</quotes>]]; F4 := [[1,<quotes>yt</quotes>], [-1, <quotes>ty</quotes>]]; G := [F1, F2,F3,F4]; NCo.IsGB(G); -- LLEX ordering (default ordering) False ------------------------------- NCo.SetOrdering(<quotes>ELIM</quotes>); NCo.IsGB(G); False -------------------------------

### See also