ApCoCoA-1:NC.HF
| This article is about a function from ApCoCoA-1. |
NC.HF
Enumerate the values of the Hilbert-Dehn function of a finitely generated K-algebra.
Syntax
NC.HF(G:LIST[, DB:INT]):LIST
Description
Let P be a finitely generated non-commutative polynomial ring over K, and let I be a finitely generated two-sided ideal in P. Then P/I is a finitely generated K-algebra. Moreover, for every integer i, we let F_{i} be the K-vector subspace generated by the words of length less than or equal to i. Clearly, the set {F_{i}} is a filtration of P. Further, the filtration {F_{i}} induces a filtration {F_{i}/(F_{i} intersects I)} of P/I. The Hilbert-Dehn function of K-algebra P/I is a map HF: N --> N defined by HF(i)=dim(F_{i}/(F_{i} intersects I))-dim(F_{i-1}/(F_{i-1} intersects I)), i.e. HF(i) is equal to the number of words of length i in a Macaulay's basis (see NC.MB) of P/I.
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 non-negative integers, which are values of the Hilbert-Dehn function of the K-algebra P/<G>.
Optional parameter:
@param DB: a positive INT, which is a degree bound of the Hilbert-Dehn function. Note that we set DB=32 by default. Thus, in the case that the K-dimension of P/<G> is finite, it is necessary to set DB to a large enough INT in order to compute all the values of the Hilbert-Dehn function.
Example
Use ZZ/(2)[t,x,y]; NC.SetOrdering(<quotes>LLEX</quotes>); 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.HF(Gb,5); [1, 3, 5, 5, 5, 5] -------------------------------
See also
