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

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

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

− | + | Let <tt>P</tt> be a finitely generated non-commutative polynomial ring over <tt>K</tt>, and let <tt>I</tt> be a finitely generated two-sided ideal in <tt>P</tt>. Then <tt>P/I</tt> is a finitely generated <tt>K</tt>-algebra. Moreover, for every integer <tt>i</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>. Clearly, the set <tt>{F_{i}}</tt> is a filtration of <tt>P</tt>. Further, the filtration <tt>{F_{i}}</tt> induces a filtration <tt>{F_{i}/(F_{i} intersects I)}</tt> of <tt>P/I</tt>. The <em>Hilbert function</em> of <tt>K</tt>-algebra <tt>P/I</tt> is a map <tt>HF: N --> N</tt> defined by <tt>HF(i)=dim(F_{i}/(F_{i} intersects I))-dim(F_{i-1}/(F_{i-1} intersects I))</tt>, i.e. <tt>HF(i)</tt> is equal to the number of words of length <tt>i</tt> in a Macaulay's basis (see <ref>NC.MB</ref>) of <tt>P/I</tt>. | |

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

<syntax> | <syntax> | ||

− | NC.HF( | + | NC.HF(G:LIST[, DB:INT]):LIST |

− | |||

</syntax> | </syntax> | ||

<description> | <description> | ||

<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | <em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | ||

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

− | Please set ring | + | Please set non-commutative polynomial ring (via the command <ref>Use</ref>) and word ordering (via the function <ref>NC.SetOrdering</ref>) before calling this function. The default word ordering is the length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant commands and functions. |

<itemize> | <itemize> | ||

− | <item>@param <em> | + | <item>@param <em>G:</em> a LIST of non-zero non-commutative polynomials, which form a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial <tt>f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5</tt> is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial <tt>0</tt> is represented as the empty LIST []. <em>Warning:</em> users should take responsibility to ensure that G is indeed a Groebner basis with respect to a length compatible word ordering!</item> |

− | <item>@param <em> | + | <item>@return: a LIST of non-negative integers, which are values of the Hilbert function of the K-algebra <tt>P<Gb></tt>.</item> |

− | + | </itemize> | |

+ | Optional parameter: | ||

+ | <itemize> | ||

+ | <item>@param <em>DB:</em> a positive INT, which is a degree bound of the Hilbert function. <em>Note that</em> we set <tt>DB=32</tt> by default. Thus, in the case that the <tt>K</tt>-dimension of <tt>P<Gb></tt> is finite, it is necessary to set <tt>DB</tt> to a large enough INT in order to compute all the values of the Hilbert function.</item> | ||

</itemize> | </itemize> | ||

<example> | <example> | ||

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</description> | </description> | ||

<seealso> | <seealso> | ||

+ | <see>Use</see> | ||

+ | <see>NC.IsGB</see> | ||

+ | <see>NC.MB</see> | ||

<see>NC.SetOrdering</see> | <see>NC.SetOrdering</see> | ||

<see>Introduction to CoCoAServer</see> | <see>Introduction to CoCoAServer</see> | ||

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<types> | <types> | ||

<type>apcocoaserver</type> | <type>apcocoaserver</type> | ||

+ | <type>ideal</type> | ||

<type>groebner</type> | <type>groebner</type> | ||

− | |||

<type>non_commutative</type> | <type>non_commutative</type> | ||

</types> | </types> |

## Revision as of 13:33, 29 April 2013

## NC.HF

Enumerate the values of the Hilbert function of a finitely generated `K`-algebra.

Let `P` be a finitely generated non-commutative polynomial ring over `K`, and let `I` be a finitely generated two-sided ideal in `P`. Then `P/I` is a finitely generated `K`-algebra. Moreover, for every integer `i`, we let `F_{i}` be the `K`-vector subspace generated by the words of length less than or equal to `i`. Clearly, the set `{F_{i}}` is a filtration of `P`. Further, the filtration `{F_{i}}` induces a filtration `{F_{i}/(F_{i} intersects I)}` of `P/I`. The *Hilbert function* of `K`-algebra `P/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 NC.MB) of `P/I`.

### Syntax

NC.HF(G: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 non-commutative polynomial ring (via the command Use) and word ordering (via the function NC.SetOrdering) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions.

@param

*G:*a LIST of non-zero non-commutative polynomials, which form a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial`f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5`is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial`0`is represented as the empty LIST [].*Warning:*users should take responsibility to ensure that G is indeed a Groebner basis with respect to a length compatible word ordering!@return: a LIST of non-negative integers, which are values of the Hilbert function of the K-algebra

`P<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`P<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

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