up previous next
NC.GHF

Compute the values of the Hilbert function of a finitely generated F2-algebra.

For every i in N, we let F_{i} be the F2-vector subspace generated by the words of length less than or equal to i. Then {F_{i}} is a filtration of F2. Let I be an ideal of F2. The filtration {F_{i}} induces a filtration {F_{i}/(F_{i} intersect I)} of F2/I. The Hilbert function of F2-algebra F2/I is a map from N to N defined by mapping i to dim(F_{i}/(F_{i} intersect I))-dim(F_{i-1}/(F_{i-1} intersect I)).
Syntax
          
NC.GHF(Gb:LIST):LIST
NC.GHF(Gb:LIST, DegreeBound:INT):LIST

          

Description
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 admissible ordering via the functions NC.SetX and NC.SetOrdering, respectively, before calling the function. The default ordering is length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

Example
NC.SetX("ab");
NC.SetOrdering("LLEX");
F1:=["aa", ""];
F2:=["bbb", ""];
F3:=["abababbabbabababbabb", ""];
G:=[F1,F2,F3]; -- a generalized triangle group 
NC.GHF(G); -- G is a partial Groeber basis!!

[1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 464, 614, 812, 
1074, 1421, 1879, 2486, 3288, 4349, 5753, 7610, 10067, 13317]
-------------------------------
GB:=NC.GGB(G);
NC.GHF(GB);

[1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 24, 27, 31, 35, 37, 38, 41, 43, 41, 34, 30, 32, 31, 23, 15, 9, 4, 1, 0]
-------------------------------
RGB:=NC.GReducedGB(G);
NC.GHF(RGB);

[1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 24, 27, 31, 35, 37, 38, 41, 43, 41, 34, 30, 32, 31, 23, 15, 9, 4, 1, 0]
-------------------------------
Sum(It); -- the order of the generalized triangle group 

576
-------------------------------


See Also