# Difference between revisions of "ApCoCoA-1:NC.HF"

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<title>NC.HF</title> | <title>NC.HF</title> | ||

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− | Hilbert function of a <tt>K</tt>-algebra. | + | Compute the values of the Hilbert function of a finitely generated <tt>K</tt>-algebra. |

+ | <par/> | ||

+ | For every <tt>i</tt> in <tt>N</tt>, we let <tt>F_{i}</tt> be the <tt>K</tt>-vector subspace generated by the words of length less than or equal to <tt>i</tt>. Then <tt>{F_{i}}</tt> is a filtration of <tt>K<X></tt>. Let <tt>I</tt> be an ideal of <tt>K<X></tt>. The filtration <tt>{F_{i}}</tt> induces a filtration <tt>{F_{i}/(F_{i} intersect I)}</tt> of <tt>K<X>/I</tt>. The <em>Hilbert function</em> of <tt>K</tt>-algebra <tt>K<X>/I</tt> is a map from <tt>N</tt> to <tt>N</tt> defined by mapping <tt>i</tt> to <tt>dim(F_{i}/(F_{i} intersect I))-dim(F_{i-1}/(F_{i-1} intersect I))</tt>. | ||

</short_description> | </short_description> | ||

<syntax> | <syntax> |

## Revision as of 15:33, 11 June 2012

## NC.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

NC.HF(Gb:LIST):LIST NC.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 NC.SetFp, NC.SetX and NC.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

NC.SetX(<quotes>xyzt</quotes>); NC.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>]]]; NC.HF(Gb, 5); [1, 4, 12, 34, 100, 292] -------------------------------

### See also