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NC.MRHF

Hilbert function of a K-algebra.

For every 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. Let I be an ideal of K. The filtration {F_{i}} induces a filtration {F_{i}/(F_{i} intersect I)} of K/I. The Hilbert function of K-algebra K/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.MRHF(X:STRING, Ordering:STRING, Relations:LIST, Gb:LIST):LIST
NC.MRHF(X:STRING, Ordering:STRING, Relations:LIST, 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 coefficient field via the function NC.SetFp (or NC.UnsetFp) before calling this function. The default coefficient field is Q. For more information, please check the relevant function.

Example
X := "xyzt"; 
Ordering := "LLEX"; 
Relations:=[]; 
F1 := [[1,"xx"], [-1,"yx"]];   
F2 := [[1,"xy"], [-1,"ty"]];  
F3 := [[1,"xt"], [-1, "tx"]];  
F4 := [[1,"yt"], [-1, "ty"]];  
G := [F1, F2,F3,F4]; 
NC.MRHF(X, Ordering, Relations, G, 5); --G is a partical Groebner basis

[1, 4, 12, 36, 108, 324]
-------------------------------
Gb:= [[[1, "yt"], [-1, "ty"]], [[1, "xt"], [-1, "tx"]], [[1, "xy"], [-1, "ty"]], [[1, "xx"], [-1, "yx"]],  
[[1, "tyy"], [-1, "tty"]], [[1, "yyx"], [-1, "tyx"]]]; 
NC.MRHF(X, Ordering, Relations, Gb, 5); --Gb is a Groeber basis (w.r.t. LLEX) of the two-sided ideal generated by G

[1, 4, 12, 34, 100, 292]
-------------------------------


See Also