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

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<command>
 
<command>
 
<title>NCo.HF</title>
 
<title>NCo.HF</title>
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</syntax>
 
</syntax>
 
<description>
 
<description>
Let <tt>K&lt;X&gt;</tt> be a finitely generated non-commutative polynomial ring, and let <tt>I</tt> be a finitely generated two-sided ideal in <tt>K&lt;X&gt;</tt>. Then <tt>K&lt;X&gt;/I</tt> is a finitely generated <tt>I</tt>-algebra. For every integer <tt>i</tt> in <tt>N</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>. Then <tt>{F_{i}}</tt> is a filtration of <tt>K&lt;X&gt;</tt>. Further, the filtration <tt>{F_{i}}</tt> induces a filtration <tt>{F_{i}/(F_{i} intersects I)}</tt> of <tt>K&lt;X&gt;/I</tt>. The <em>Hilbert function</em> of <tt>K&lt;X&gt;/I</tt> is a map <tt>HF: N --&gt;</tt> N 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>NCo.MB</ref>) of <tt>K&lt;X&gt;/I</tt>.
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Let <tt>K&lt;X&gt;</tt> be a finitely generated non-commutative polynomial ring, and let <tt>I</tt> be a finitely generated two-sided ideal in <tt>K&lt;X&gt;</tt>. Then <tt>K&lt;X&gt;/I</tt> is a finitely generated <tt>I</tt>-algebra. For every integer <tt>i</tt> in <tt>N</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>. Then <tt>{F_{i}}</tt> is a filtration of <tt>K&lt;X&gt;</tt>. Further, the filtration <tt>{F_{i}}</tt> induces a filtration <tt>{F_{i}/(F_{i} intersects I)}</tt> of <tt>K&lt;X&gt;/I</tt>. The <em>Hilbert function</em> of <tt>K&lt;X&gt;/I</tt> is a map <tt>HF: N --&gt;</tt> N 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:NCo.MB|NCo.MB</ref>) of <tt>K&lt;X&gt;/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>coefficient field</em> <tt> K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>NCo.SetFp</ref>, <ref>NCo.SetX</ref> and <ref>NCo.SetOrdering</ref>, respectively, before calling this function. The default coefficient field is <tt>Q</tt>, and 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>coefficient field</em> <tt> K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>ApCoCoA-1:NCo.SetFp|NCo.SetFp</ref>, <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref>, respectively, before calling this function. The default coefficient field is <tt>Q</tt>, and 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 <tt>K&lt;X&gt;</tt> forming a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <tt>&lt;X&gt;</tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <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 <tt>K&lt;X&gt;</tt> forming a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <tt>&lt;X&gt;</tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. <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>K</tt>-algebra <tt>K&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>K</tt>-algebra <tt>K&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:= [[[1, "yt"], [-1, "ty"]], [[1, "xt"], [-1, "tx"]], [[1, "xy"], [-1, "ty"]], [[1, "xx"], [-1, "yx"]],   
[[1, <quotes>tyy</quotes>], [-1, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [-1, <quotes>tyx</quotes>]]];  
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[[1, "tyy"], [-1, "tty"]], [[1, "yyx"], [-1, "tyx"]]];  
 
NCo.HF(Gb, 5);
 
NCo.HF(Gb, 5);
 
[1, 4, 12, 34, 100, 292]
 
[1, 4, 12, 34, 100, 292]
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</description>
 
</description>
 
<seealso>
 
<seealso>
<see>NCo.IsGB</see>
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<see>ApCoCoA-1:NCo.IsGB|NCo.IsGB</see>
<see>NCo.MB</see>
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<see>ApCoCoA-1:NCo.MB|NCo.MB</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.HF</key>
 
<key>NCo.HF</key>
 
<key>HF</key>
 
<key>HF</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 13:39, 29 October 2020

This article is about a function from ApCoCoA-1.

NCo.HF

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

Syntax

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

Description

Let K<X> be a finitely generated non-commutative polynomial ring, and let I be a finitely generated two-sided ideal in K<X>. Then K<X>/I is a finitely generated I-algebra. For every integer i in N, we let F_{i} be the K-vector subspace generated by the words of length less than or equal to i. Then {F_{i}} is a filtration of K<X>. Further, the filtration {F_{i}} induces a filtration {F_{i}/(F_{i} intersects I)} of K<X>/I. The Hilbert function of K<X>/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 NCo.MB) of K<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 coefficient field K, alphabet (or set of indeterminates) X and ordering via the functions NCo.SetFp, NCo.SetX and NCo.SetOrdering, respectively, before calling this function. The default coefficient field is Q, and 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 K<X> forming a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <X> and C is the coefficient of W. For example, the polynomial f=xy-y+1 is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. 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 K-algebra K<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 K-dimension of K<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:= [[[1, "yt"], [-1, "ty"]], [[1, "xt"], [-1, "tx"]], [[1, "xy"], [-1, "ty"]], [[1, "xx"], [-1, "yx"]],  
[[1, "tyy"], [-1, "tty"]], [[1, "yyx"], [-1, "tyx"]]]; 
NCo.HF(Gb, 5);
[1, 4, 12, 34, 100, 292]
-------------------------------

See also

NCo.IsGB

NCo.MB

NCo.SetFp

NCo.SetOrdering

NCo.SetX

Introduction to CoCoAServer