# Difference between revisions of "ApCoCoA-1:NCo.BHF"

(New page: <command> <title>NCo.BHF</title> <short_description> Enumerate values of the Hilbert function of a finitely generated algebra over the binary field F_{2}={0,1}. <par/> Let <tt>F_{2}<X&g...) |
|||

Line 4: | Line 4: | ||

Enumerate values of the Hilbert function of a finitely generated algebra over the binary field F_{2}={0,1}. | Enumerate values of the Hilbert function of a finitely generated algebra over the binary field F_{2}={0,1}. | ||

<par/> | <par/> | ||

− | Let <tt>F_{2}<X></tt> be a finitely generated free monoid ring, and let <tt>I</tt> be a finitely generated two-sided ideal in <tt>F_{2}<X></tt>. Then <tt>F_{2}<X>/I</tt> is a finitely generated <tt> | + | Let <tt>F_{2}<X></tt> be a finitely generated free monoid ring, and let <tt>I</tt> be a finitely generated two-sided ideal in <tt>F_{2}<X></tt>. Then <tt>F_{2}<X>/I</tt> is a finitely generated <tt>F_{2}</tt>-algebra. For every integer <tt>i</tt> in <tt>N</tt>, we let <tt>F_{i}</tt> be the <tt>F_{2}</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>F_{2}<X></tt>. Further, the filtration <tt>{F_{i}}</tt> induces a filtration <tt>{F_{i}/(F_{i} intersects I)}</tt> of <tt>F_{2}<X>/I</tt>. The <em>Hilbert function</em> of <tt>F_{2}<X>/I</tt> is a map <tt>BHF: N --></tt> N defined by <tt>BHF(i)=dim(F_{i}/(F_{i} intersects I))-dim(F_{i-1}/(F_{i-1} intersects I))</tt>, i.e. <tt>BHF(i)</tt> is equal to the number of words of length <tt>i</tt> in a Macaulay's basis (see <ref>NCo.BMB</ref>) of <tt>F_{2}<X>/I</tt>. |

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

<syntax> | <syntax> |

## Revision as of 16:26, 1 May 2013

## NCo.BHF

Enumerate values of the Hilbert function of a finitely generated algebra over the binary field F_{2}={0,1}.

Let `F_{2}<X>` be a finitely generated free monoid ring, and let `I` be a finitely generated two-sided ideal in `F_{2}<X>`. Then `F_{2}<X>/I` is a finitely generated `F_{2}`-algebra. For every integer `i` in `N`, we let `F_{i}` be the `F_{2}`-vector subspace generated by the words of length less than or equal to `i`. Then `{F_{i}}` is a filtration of `F_{2}<X>`. Further, the filtration `{F_{i}}` induces a filtration `{F_{i}/(F_{i} intersects I)}` of `F_{2}<X>/I`. The *Hilbert function* of `F_{2}<X>/I` is a map `BHF: N -->` N defined by `BHF(i)=dim(F_{i}/(F_{i} intersects I))-dim(F_{i-1}/(F_{i-1} intersects I))`, i.e. `BHF(i)` is equal to the number of words of length `i` in a Macaulay's basis (see NCo.BMB) of `F_{2}<X>/I`.

### Syntax

NCo.BHF(Gb:LIST[, DB: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 *alphabet* (or set of indeterminates) `X` and *word ordering* via the functions NCo.SetX and NCo.SetOrdering, respectively, before calling this function. 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 the free monoid ring`F_{2}<X>`which is a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of words (or terms) in`<X>`. Each word is represented as a STRING. For example,`xy^2x`is represented as "xyyx", and the identity is represented as the empty string "". Thus, the polynomial`f=xy-y+1`is represented as F:=["xy", "y", ""]. The zero polynomial`0`is represented as the empty LIST [].*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

`F_{2}`-algebra`F_{2}<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`F_{2}`-dimension of`F_{2}<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.BHF(Gb, 5); [1, 4, 12, 34, 100, 292] -------------------------------

### See also