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

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{{Version|1}}
 
<command>
 
<command>
 
<title>NCo.BHF</title>
 
<title>NCo.BHF</title>
 
<short_description>
 
<short_description>
 
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/>
 
Let <tt>F_{2}&lt;X&gt;</tt> be a finitely generated free monoid ring, and let <tt>I</tt> be a finitely generated two-sided ideal in <tt>F_{2}&lt;X&gt;</tt>. Then <tt>F_{2}&lt;X&gt;/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}&lt;X&gt;</tt>. Further, the filtration <tt>{F_{i}}</tt> induces a filtration <tt>{F_{i}/(F_{i} intersects I)}</tt> of <tt>F_{2}&lt;X&gt;/I</tt>. The <em>Hilbert function</em> of <tt>F_{2}&lt;X&gt;/I</tt> is a map <tt>BHF: N --&gt;</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}&lt;X&gt;/I</tt>.
 
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
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</syntax>
 
</syntax>
 
<description>
 
<description>
 +
Let <tt>F_{2}&lt;X&gt;</tt> be a finitely generated free monoid ring, and let <tt>I</tt> be a finitely generated two-sided ideal in <tt>F_{2}&lt;X&gt;</tt>. Then <tt>F_{2}&lt;X&gt;/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}&lt;X&gt;</tt>. Further, the filtration <tt>{F_{i}}</tt> induces a filtration <tt>{F_{i}/(F_{i} intersects I)}</tt> of <tt>F_{2}&lt;X&gt;/I</tt>. The <em>Hilbert function</em> of <tt>F_{2}&lt;X&gt;/I</tt> is a map <tt>BHF: N --&gt;</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}&lt;X&gt;/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 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 (<quotes>LLEX</quotes>). For more information, please check the relevant functions.
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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}&lt;X&gt;</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>&lt;X&gt;</tt>. Each word is represented as a STRING. For example, <tt>xy^2x</tt> is represented as <quotes>xyyx</quotes>, and the identity is represented as the empty string <quotes></quotes>. Thus, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[<quotes>xy</quotes>, <quotes>y</quotes>, <quotes></quotes>]. 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>
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<item>@param <em>Gb:</em> a LIST of non-zero polynomials in the free monoid ring <tt>F_{2}&lt;X&gt;</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>&lt;X&gt;</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}&lt;X&gt;/&lt;Gb&gt;</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}&lt;X&gt;/&lt;Gb&gt;</tt>.</item>
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</itemize>
 
</itemize>
 
<example>
 
<example>
NCo.SetX(<quotes>xyzt</quotes>);  
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NCo.SetX("xyzt");  
NCo.SetOrdering(<quotes>LLEX</quotes>);  
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NCo.SetOrdering("LLEX");  
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>]],
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Gb:= [["yt", "ty"], ["xt", "tx"], ["xy", "ty"], ["xx", "yx"], ["tyy", "tty"], ["yyx", "tyx"]];
[[1, <quotes>tyy</quotes>], [-1, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [-1, <quotes>tyx</quotes>]]];  
 
 
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>
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<see>ApCoCoA-1:NCo.BIsGB|NCo.BIsGB</see>
<see>NCo.BMB</see>
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<see>ApCoCoA-1:NCo.BMB|NCo.BMB</see>
<see>NCo.SetFp</see>
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<see>ApCoCoA-1:NCo.SetFp|NCo.SetFp</see>
<see>NCo.SetOrdering</see>
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<see>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</see>
<see>NCo.SetX</see>
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<see>ApCoCoA-1:NCo.SetX|NCo.SetX</see>
<see>Introduction to CoCoAServer</see>
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<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>
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<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

NCo.BIsGB

NCo.BMB

NCo.SetFp

NCo.SetOrdering

NCo.SetX

Introduction to CoCoAServer