# ApCoCoA-1:NC.IsGB

## NC.IsGB

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

### Syntax

NC.IsGB(G:LIST):BOOL

### Description

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 `LW{G}` generates the leading word ideal `LW(<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`.

*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

Use ZZ/(2)[t,x,y]; G := [[[x^2], [y, x]], [[t, y], [x, y]], [[y, t], [x, y]], [[t, x], [x, t]], [[x, y, x], [y^2, x]], [[x, y^2], [y, x, y]], [[y, x, t], [y^2, x]]]; NC.SetOrdering(<quotes>ELIM</quotes>); NC.IsGB(G); True ------------------------------- NC.SetOrdering(<quotes>LLEX</quotes>); NC.IsGB(G); False -------------------------------

### See also