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NCo.HF

Enumerate values of the Hilbert-Dehn function of a finitely generated K-algebra.
Syntax
          
NCo.HF(Gb:LIST[, DB:INT]):LIST

          

Description
Let K be a finitely generated non-commutative polynomial ring, and let I be a finitely generated two-sided ideal in K. Then K/I is a finitely generated I-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 K/I. The Hilbert function of K/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.MB) of K/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 coefficient field K, alphabet (or set of indeterminates) X and ordering via the functions NCo.SetFp, NCo.SetX and NCo.SetOrdering, respectively, before calling this function. The default coefficient field is Q, and 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:= [[[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.HF(Gb, 5);
[1, 4, 12, 34, 100, 292]
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See Also