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

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<command> | <command> | ||

<title>NCo.BHF</title> | <title>NCo.BHF</title> | ||

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

<description> | <description> | ||

− | 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>. | + | 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>ApCoCoA-1:NCo.BMB|NCo.BMB</ref>) of <tt>F_{2}<X>/I</tt>. |

<par/> | <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 ring environment <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>word ordering</em> via the functions <ref>NCo.SetX</ref> and <ref>NCo.SetOrdering</ref>, respectively, before calling this function. The default ordering is the length-lexicographic ordering ( | + | Please set ring environment <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>word ordering</em> via the functions <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref>, respectively, before calling this function. The default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions. |

<itemize> | <itemize> | ||

− | <item>@param <em>Gb:</em> a LIST of non-zero polynomials in the free monoid ring <tt>F_{2}<X></tt> 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 <tt><X></tt>. Each word is represented as a STRING. For example, <tt>xy^2x</tt> is represented as | + | <item>@param <em>Gb:</em> a LIST of non-zero polynomials in the free monoid ring <tt>F_{2}<X></tt> 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 <tt><X></tt>. Each word is represented as a STRING. For example, <tt>xy^2x</tt> is represented as "xyyx", and the identity is represented as the empty string "". Thus, the polynomial <tt>f=xy-y+1</tt> is represented as F:=["xy", "y", ""]. The zero polynomial <tt>0</tt> is represented as the empty LIST []. <em>Warning:</em> 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.</item> |

<item>@return: a LIST of non-negative integers, which are values of the Hilbert function of the <tt>F_{2}</tt>-algebra <tt>F_{2}<X>/<Gb></tt>.</item> | <item>@return: a LIST of non-negative integers, which are values of the Hilbert function of the <tt>F_{2}</tt>-algebra <tt>F_{2}<X>/<Gb></tt>.</item> | ||

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

<example> | <example> | ||

− | NCo.SetX( | + | NCo.SetX("xyzt"); |

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

− | Gb:= [[ | + | Gb:= [["yt", "ty"], ["xt", "tx"], ["xy", "ty"], ["xx", "yx"], ["tyy", "tty"], ["yyx", "tyx"]]; |

NCo.BHF(Gb, 5); | NCo.BHF(Gb, 5); | ||

[1, 4, 12, 34, 100, 292] | [1, 4, 12, 34, 100, 292] | ||

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

<seealso> | <seealso> | ||

− | <see>NCo.BIsGB</see> | + | <see>ApCoCoA-1:NCo.BIsGB|NCo.BIsGB</see> |

− | <see>NCo.BMB</see> | + | <see>ApCoCoA-1:NCo.BMB|NCo.BMB</see> |

− | <see>NCo.SetFp</see> | + | <see>ApCoCoA-1:NCo.SetFp|NCo.SetFp</see> |

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

− | <see>NCo.SetX</see> | + | <see>ApCoCoA-1:NCo.SetX|NCo.SetX</see> |

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

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

<types> | <types> | ||

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<key>NCo.BHF</key> | <key>NCo.BHF</key> | ||

<key>BHF</key> | <key>BHF</key> | ||

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

</command> | </command> |

## Latest revision as of 13:37, 29 October 2020

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

## NCo.BHF

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

### Syntax

NCo.BHF(Gb:LIST[, DB:INT]):LIST

### Description

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`.

*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("xyzt"); NCo.SetOrdering("LLEX"); Gb:= [["yt", "ty"], ["xt", "tx"], ["xy", "ty"], ["xx", "yx"], ["tyy", "tty"], ["yyx", "tyx"]]; NCo.BHF(Gb, 5); [1, 4, 12, 34, 100, 292] -------------------------------

### See also