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

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Check whether a LIST of non-zero polynomials is a Groebner basis in a non-commutative polynomial ring. | Check whether a LIST of non-zero polynomials is a Groebner basis in a non-commutative polynomial ring. | ||

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

− | Note that, given | + | Note that, given a word ordering, a set of non-zero polynomials <tt>G</tt> is called a <em>Groebner basis</em> of with respect to this ordering if the leading word set <tt>LT{G}</tt> generates the leading word ideal <tt>LT(<G>)</tt>. This function checks whether a given finite set of non-zero polynomial <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 <tt>S-polynomials</tt> of all obstructions of <tt>G</tt> 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 | + | Please set non-commutative polynomial ring (via the command <ref>Use</ref>) and word ordering (via the function <ref>NC.SetOrdering</ref>) before calling this function. The default word ordering is the length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant commands and functions. |

<itemize> | <itemize> | ||

− | <item>@param <em>G</em>: a LIST of non-zero polynomials | + | <item>@param <em>G</em>: a LIST of non-zero non-commutative polynomials. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial <tt>f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5</tt> is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial <tt>0</tt> is represented as the empty LIST [].</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 ordering and False otherwise.</item> |

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

<example> | <example> | ||

Line 36: | Line 36: | ||

</description> | </description> | ||

<seealso> | <seealso> | ||

+ | <see>Use</see> | ||

<see>NC.GB</see> | <see>NC.GB</see> | ||

<see>NC.ReducedGB</see> | <see>NC.ReducedGB</see> |

## Revision as of 12:43, 26 April 2013

## NC.IsGB

Check whether a LIST of non-zero polynomials is a Groebner basis in a non-commutative polynomial ring.

Note that, given a word ordering, a set of non-zero polynomials `G` is called a *Groebner basis* of with respect to this ordering if the leading word set `LT{G}` generates the leading word ideal `LT(<G>)`. This function checks whether a given finite set of non-zero polynomial `G` is a Groebner basis by using the `Buchberger Criterion`, i.e. `G` is a Groebner basis if the `S-polynomials` of all obstructions of `G` have the zero normal remainder with respect to `G`.

### Syntax

NC.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 non-commutative polynomial ring (via the command Use) and word ordering (via the function NC.SetOrdering) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions.

@param

*G*: a LIST of non-zero non-commutative polynomials. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial`f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5`is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial`0`is represented as the empty LIST [].@return: a BOOL, which is True if

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

#### Example

NC.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]; NC.IsGB(G); -- LLEX ordering (default ordering) False ------------------------------- NC.SetOrdering(<quotes>ELIM</quotes>); NC.IsGB(G); False -------------------------------

### See also