ApCoCoA-1:NCo.BHF

This article is about a function from ApCoCoA-1.

NCo.BHF

Enumerate values of the Hilbert function of a finitely generated algebra over the binary field F_{2}={0,1}.

Syntax

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

Description

Let F_{2}<X> be a finitely generated free monoid ring, and let I be a finitely generated two-sided ideal in F_{2}<X>. Then F_{2}<X>/I is a finitely generated F_{2}-algebra. For every integer i in N, we let F_{i} be the F_{2}-vector subspace generated by the words of length less than or equal to i. Then {F_{i}} is a filtration of F_{2}<X>. Further, the filtration {F_{i}} induces a filtration {F_{i}/(F_{i} intersects I)} of F_{2}<X>/I. The Hilbert function of F_{2}<X>/I is a map BHF: N --> N defined by BHF(i)=dim(F_{i}/(F_{i} intersects I))-dim(F_{i-1}/(F_{i-1} intersects I)), i.e. BHF(i) is equal to the number of words of length i in a Macaulay's basis (see NCo.BMB) of F_{2}<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 alphabet (or set of indeterminates) X and word ordering via the functions NCo.SetX and NCo.SetOrdering, respectively, before calling this function. 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 the free monoid ring F_{2}<X> which is a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of words (or terms) in <X>. Each word is represented as a STRING. For example, xy^2x is represented as "xyyx", and the identity is represented as the empty string "". Thus, the polynomial f=xy-y+1 is represented as F:=["xy", "y", ""]. The zero polynomial 0 is represented as the empty LIST []. 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 F_{2}-algebra F_{2}<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 F_{2}-dimension of F_{2}<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("xyzt"); 
NCo.SetOrdering("LLEX"); 
Gb:= [["yt", "ty"], ["xt", "tx"], ["xy", "ty"], ["xx", "yx"], ["tyy", "tty"], ["yyx", "tyx"]];
NCo.BHF(Gb, 5);
[1, 4, 12, 34, 100, 292]
-------------------------------

See also

NCo.BIsGB

NCo.BMB

NCo.SetFp

NCo.SetOrdering

NCo.SetX

Introduction to CoCoAServer