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NCo.MRHF

Enumerate values of the Hilbert function of a finitely generated K-algebra.
Syntax
          
NCo.MRHF(X:STRING, Ordering:STRING, Relations:LIST, G:LIST[, DB:INT]):LIST

          

Description
Let P=K be a finitely generated monoid ring, and let I be a finitely generated two-sided ideal in P. Then P/I is a finitely generated K-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. Further, the filtration {F_{i}} induces a filtration {F_{i}/(F_{i} intersects I)} of P/I. The Hilbert function of P/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.MRMB) of P/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 coefficient field via the function NCo.SetFp (or NCo.UnsetFp) before calling this function. The default coefficient field is the field of rational numbers, i.e. RAT in CoCoAL. For more information, please check the relevant functions. Optional parameter:

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]; 
NCo.MRHF(X, Ordering, Relations, G, 5); --G is a partical Groebner basis

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

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


See Also