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

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+ | {{Version|1}} | ||

<command> | <command> | ||

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

<short_description> | <short_description> | ||

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

− | |||

− | |||

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

<syntax> | <syntax> | ||

Line 10: | Line 9: | ||

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

<description> | <description> | ||

+ | 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-Dehn 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>ApCoCoA-1:NC.MB|NC.MB</ref>) of <tt>P/I</tt>. | ||

+ | <par/> | ||

<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 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 ( | + | Please set non-commutative polynomial ring (via the command <ref>ApCoCoA-1:Use|Use</ref>) and word ordering (via the function <ref>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</ref>) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions. |

<itemize> | <itemize> | ||

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

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

Optional parameter: | Optional parameter: | ||

<itemize> | <itemize> | ||

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

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

<example> | <example> | ||

− | + | Use ZZ/(2)[t,x,y]; | |

− | NC.SetOrdering( | + | NC.SetOrdering("LLEX"); |

− | + | F1 := [[x^2], [y,x]]; -- x^2+yx | |

− | [[ | + | F2 := [[x,y], [t,y]]; -- xy+ty |

− | NC.HF(Gb, 5); | + | F3 := [[x,t], [t,x]]; -- xt+tx |

− | [1, | + | F4 := [[y,t], [t,y]]; -- yt+ty |

+ | G := [F1, F2,F3,F4]; | ||

+ | Gb:=NC.GB(G); | ||

+ | NC.HF(Gb,5); | ||

+ | |||

+ | [1, 3, 5, 5, 5, 5] | ||

------------------------------- | ------------------------------- | ||

</example> | </example> | ||

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

<seealso> | <seealso> | ||

− | <see>Use</see> | + | <see>ApCoCoA-1:Use|Use</see> |

− | <see>NC.IsGB</see> | + | <see>ApCoCoA-1:NC.IsGB|NC.IsGB</see> |

− | <see>NC.MB</see> | + | <see>ApCoCoA-1:NC.MB|NC.MB</see> |

− | <see>NC.SetOrdering</see> | + | <see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see> |

− | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> |

</seealso> | </seealso> | ||

<types> | <types> | ||

Line 47: | Line 53: | ||

<key>NC.HF</key> | <key>NC.HF</key> | ||

<key>HF</key> | <key>HF</key> | ||

− | <wiki-category>Package_ncpoly</wiki-category> | + | <wiki-category>ApCoCoA-1:Package_ncpoly</wiki-category> |

</command> | </command> |

## Latest revision as of 13:34, 29 October 2020

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

## NC.HF

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

### Syntax

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

### Description

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-Dehn 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`.

*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-Dehn function of the K-algebra

`P/<G>`.

Optional parameter:

@param

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

#### Example

Use ZZ/(2)[t,x,y]; NC.SetOrdering("LLEX"); F1 := [[x^2], [y,x]]; -- x^2+yx F2 := [[x,y], [t,y]]; -- xy+ty F3 := [[x,t], [t,x]]; -- xt+tx F4 := [[y,t], [t,y]]; -- yt+ty G := [F1, F2,F3,F4]; Gb:=NC.GB(G); NC.HF(Gb,5); [1, 3, 5, 5, 5, 5] -------------------------------

### See also