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

Line 3: | Line 3: | ||

<short_description> | <short_description> | ||

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

− | |||

− | |||

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

<syntax> | <syntax> | ||

Line 10: | Line 8: | ||

</syntax> | </syntax> | ||

<description> | <description> | ||

+ | 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>. | ||

+ | <par/> | ||

<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/> |

## Revision as of 19:57, 14 May 2013

## NCo.IsGB

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

### Syntax

NCo.IsGB(G:LIST):BOOL

### Description

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`.

*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