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

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<title>NC.IsGB</title> | <title>NC.IsGB</title> | ||

<short_description> | <short_description> | ||

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

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

Note that, given an ideal <tt>I</tt> and an admissible ordering <tt>Ordering</tt>, a set of non-zero polynomials <tt>Gb</tt> is called a <em>Groebner basis</em> of <tt>I</tt> w.r.t. <tt>Ordering</tt> if the leading term set <tt>LT{Gb}</tt> (w.r.t. <tt>Ordering</tt>) generates the leading term ideal <tt>LT(I)</tt> (w.r.t. <tt>Ordering</tt>). The function check 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 all the <tt>S-polynomials</tt> of obstructions have the zero normal remainder w.r.t. <tt>G</tt>. | Note that, given an ideal <tt>I</tt> and an admissible ordering <tt>Ordering</tt>, a set of non-zero polynomials <tt>Gb</tt> is called a <em>Groebner basis</em> of <tt>I</tt> w.r.t. <tt>Ordering</tt> if the leading term set <tt>LT{Gb}</tt> (w.r.t. <tt>Ordering</tt>) generates the leading term ideal <tt>LT(I)</tt> (w.r.t. <tt>Ordering</tt>). The function check 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 all the <tt>S-polynomials</tt> of obstructions have the zero normal remainder w.r.t. <tt>G</tt>. | ||

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

<seealso> | <seealso> | ||

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

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

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<see>NC.SetOrdering</see> | <see>NC.SetOrdering</see> | ||

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<see>NC.TruncatedGB</see> | <see>NC.TruncatedGB</see> | ||

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<see>Introduction to CoCoAServer</see> | <see>Introduction to CoCoAServer</see> | ||

</seealso> | </seealso> | ||

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

</types> | </types> | ||

− | <key> | + | <key>ncpoly.IsGB</key> |

<key>NC.IsGB</key> | <key>NC.IsGB</key> | ||

<key>IsGB</key> | <key>IsGB</key> | ||

− | <wiki-category> | + | <wiki-category>Package_ncpoly</wiki-category> |

</command> | </command> |

## Revision as of 16:59, 25 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 an ideal `I` and an admissible ordering `Ordering`, a set of non-zero polynomials `Gb` is called a *Groebner basis* of `I` w.r.t. `Ordering` if the leading term set `LT{Gb}` (w.r.t. `Ordering`) generates the leading term ideal `LT(I)` (w.r.t. `Ordering`). The function check 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 all the `S-polynomials` of obstructions have the zero normal remainder w.r.t. `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 ring environment *coefficient field* `K`, *alphabet* (or set of indeterminates) `X` and *ordering* via the functions NC.SetFp, NC.SetX and NC.SetOrdering, respectively, before calling the function. The default coefficient field is `Q`. The default ordering is 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 pairs 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 value which is True if

`G`is a Groebner basis w.r.t. 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