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Enumerate a Macaulay's basis of a finitely generated K-algebra.

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 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.

  • @param G: a LIST of non-zero non-commutative polynomials, which form a Groebner basis with respect to a length compatible word ordering. 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 []. Warning: users should take responsibility to ensure that G is indeed a Groebner basis with respect to a length compatible word ordering!

  • @return: a LIST of words forming a Macaulay's basis of the K-algebra P/<G>.

Optional parameter:

  • @param DB: a positive INT, which is a degree bound of the lengths of words. Note that we set DB=32 by default. Thus, in the case that P/<G> has a finite Macaulay's basis, it is necessary to set DB to a large enough INT in order to compute the whole Macaulay's basis.


Use ZZ/(2)[t,x,y];
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]; 

[[[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






Introduction to CoCoAServer