# ApCoCoA-1:NCo.HF

This article is about a function from ApCoCoA-1. |

## 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<X>` be a finitely generated non-commutative polynomial ring, and let `I` be a finitely generated two-sided ideal in `K<X>`. Then `K<X>/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<X>`. Further, the filtration `{F_{i}}` induces a filtration `{F_{i}/(F_{i} intersects I)}` of `K<X>/I`. The *Hilbert function* of `K<X>/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<X>/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.

@param

*Gb:*a LIST of non-zero polynomials in`K<X>`forming a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of monomials, which are LISTs 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 with respect to a length compatible word ordering! In the case that Gb is a partical Groebner basis, the function enumerates the values of a pseudo Hilbert function.@return: a LIST of non-negative integers, which are values of the Hilbert function of the

`K`-algebra`K<X>/<Gb>`.

Optional parameter:

@param

*DB:*a positive INT, which is a degree bound of the Hilbert function.*Note that*we set DB=32 by default. Thus, in the case that the`K`-dimension of`K<X>/<Gb>`is finite, it is necessary to set DB to a large enough INT in order to compute all the values of the Hilbert function.

#### 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] -------------------------------

### See also