ApCoCoA-1:NCo.HF

This article is about a function from ApCoCoA-1.

NCo.HF

Enumerate values of the Hilbert-Dehn function of a finitely generated K-algebra.

Syntax

NCo.HF(Gb:LIST[, DB:INT]):LIST

Description

Let K<X> be a finitely generated non-commutative polynomial ring, and let I be a finitely generated two-sided ideal in K<X>. Then K<X>/I is a finitely generated I-algebra. For every integer i in N, we let F_{i} be the K-vector subspace generated by the words of length less than or equal to i. Then {F_{i}} is a filtration of K<X>. Further, the filtration {F_{i}} induces a filtration {F_{i}/(F_{i} intersects I)} of K<X>/I. The Hilbert function of K<X>/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 NCo.MB) of K<X>/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 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 calling 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 Gb: a LIST of non-zero polynomials in K<X> forming a Groebner basis with respect to a length compatible word ordering. 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,""]]. Warning: users should take responsibility to make sure that Gb is indeed a Groebner basis with respect to a length compatible word ordering! In the case that Gb is a partical Groebner basis, the function enumerates the values of a pseudo Hilbert function.

• @return: a LIST of non-negative integers, which are values of the Hilbert function of the K-algebra K<X>/<Gb>.

Optional parameter:

• @param DB: a positive INT, which is a degree bound of the Hilbert function. Note that we set DB=32 by default. Thus, in the case that the K-dimension of K<X>/<Gb> is finite, it is necessary to set DB to a large enough INT in order to compute all the values of the Hilbert function.

Example

NCo.SetX(<quotes>xyzt</quotes>); 
NCo.SetOrdering(<quotes>LLEX</quotes>); 
Gb:= [[[1, <quotes>yt</quotes>], [-1, <quotes>ty</quotes>]], [[1, <quotes>xt</quotes>], [-1, <quotes>tx</quotes>]], [[1, <quotes>xy</quotes>], [-1, <quotes>ty</quotes>]], [[1, <quotes>xx</quotes>], [-1, <quotes>yx</quotes>]],  
[[1, <quotes>tyy</quotes>], [-1, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [-1, <quotes>tyx</quotes>]]]; 
NCo.HF(Gb, 5);
[1, 4, 12, 34, 100, 292]
-------------------------------

See also

NCo.IsGB

NCo.MB

NCo.SetFp

NCo.SetOrdering

NCo.SetX

Introduction to CoCoAServer