up previous next
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(). 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.

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("ELIM"); 
NC.IsGB(G);

True
-------------------------------
NC.SetOrdering("LLEX"); 
NC.IsGB(G);

False
-------------------------------


See Also