# ApCoCoA-1:NCo.HF

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## NCo.HF

Compute the values of the Hilbert function of a finitely generated 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<X>. Let I be an ideal of K<X>. The filtration {F_{i}} induces a filtration {F_{i}/(F_{i} intersect I)} of K<X>/I. The Hilbert function of K-algebra K<X>/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

```NCo.HF(Gb:LIST):LIST
NCo.HF(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 coefficient field K, alphabet (or set of indeterminates) X and ordering via the functions NCo.SetFp, NCo.SetX and NCo.SetOrdering, respectively, before calling the function. The default coefficient field is Q. The default ordering is length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

• @param Gb: a LIST of non-zero polynomials in K<X> which is a Groebner basis (w.r.t. a length compatible admissible ordering, say Ordering) of the two-sided ideal generated by Gb. Each polynomial is represented as a LIST of monomials, which are pairs of the form [C, W] where W is a word in <X> and C is the coefficient of W. For example, the polynomial F=xy-y+1 is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. Warning: users should take responsibility to make sure that Gb is indeed a Groebner basis w.r.t. Ordering! In the case that Gb is a partical Groebner basis, the function enumerates pseudo values.

• @param DegreeBound: (optional) a positive integer which is a degree bound of Hilbert funtion. Note that we set DegreeBound=32 by default. Thus to compute all the values of the Hilbert function, it is necessary to set DegreeBound to a larger enough number.

• @return: a LIST of non-negative integers, which is a list of values of the Hilbert function of the K-algebra K<X>/(Gb).

#### Example

```NCo.SetX(<quotes>xyzt</quotes>);
NCo.SetOrdering(<quotes>LLEX</quotes>);
Gb:= [[[1, <quotes>yt</quotes>], [-1, <quotes>ty</quotes>]], [[1, <quotes>xt</quotes>], [-1, <quotes>tx</quotes>]], [[1, <quotes>xy</quotes>], [-1, <quotes>ty</quotes>]], [[1, <quotes>xx</quotes>], [-1, <quotes>yx</quotes>]],
[[1, <quotes>tyy</quotes>], [-1, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [-1, <quotes>tyx</quotes>]]];
NCo.HF(Gb, 5);
[1, 4, 12, 34, 100, 292]
-------------------------------
```