up previous next
NC.MB

Enumerate a Macaulay's basis of a finitely generated K-algebra.
Syntax
          
NC.MB(G:LIST[, DB:INT]):LIST

          

Description
Given a two-sided ideal I in a non-commutative polynomial ring P over K, we can consider the K-algebra P/I as a K-vector space. Moreover, let G be a Groebner basis of I, and let B be the set of all words which are not a multiple of any word in the leading word set LW{G}. Then the residue class of the words in B form a K-basis, called a Macaulay's basis, of P/I. For the sake of computing the values of the Hilbert-Dehn function (see NC.HF) of P/I, in this function we require that G has to be a Groebner basis with respect to a length compatible word ordering (see NC.SetOrdering).

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. Optional parameter:

Example
Use ZZ/(2)[t,x,y];
NC.SetOrdering("LLEX"); 
F1 := [[x^2], [y,x]]; -- x^2+yx
F2 := [[x,y], [t,y]]; -- xy+ty
F3 := [[x,t], [t,x]]; -- xt+tx
F4 := [[y,t], [t,y]]; -- yt+ty
G := [F1, F2,F3,F4]; 
Gb:=NC.GB(G);
NC.MB(Gb,3);

[[[1]], [[y], [x], [t]], [[y, t], [y, x], [y^2], [x, t], [t^2]], 
[[y^3], [y^2, x], [y^2, t], [x, t^2], [t^3]]]
-------------------------------


See Also