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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} be a finitely generated free monoid ring, and let I be a finitely generated two-sided ideal in F_{2}. Then F_{2}/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}. Further, the filtration {F_{i}} induces a filtration {F_{i}/(F_{i} intersects I)} of F_{2}/I. The Hilbert function of F_{2}/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}/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. Optional parameter:

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]
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See Also