ApCoCoA-1:NCo.IsGB

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

NCo.GB

NCo.LW

NCo.ReducedGB

NCo.SetFp

NCo.SetOrdering

NCo.SetX

NCo.TruncatedGB

Introduction to CoCoAServer