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

## NC.IsGB

Check whether a finite list (set) of non-zero polynomials in a free monoid ring is a Groebner basis.

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
-------------------------------
```