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

From ApCoCoAWiki
(New page: <command> <title>NCo.HF</title> <short_description> Compute the values of the Hilbert function of a finitely generated <tt>K</tt>-algebra. <par/> For every <tt>i</tt> in <tt>N</tt>, we let...)
 
Line 2: Line 2:
 
<title>NCo.HF</title>
 
<title>NCo.HF</title>
 
<short_description>
 
<short_description>
−
Compute the values of the Hilbert function of a finitely generated <tt>K</tt>-algebra.
+
Enumerate values of the Hilbert function of a finitely generated <tt>K</tt>-algebra.
 
<par/>
 
<par/>
−
For every <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>. Let <tt>I</tt> be an ideal of <tt>K&lt;X&gt;</tt>. The filtration <tt>{F_{i}}</tt> induces a filtration <tt>{F_{i}/(F_{i} intersect I)}</tt> of <tt>K&lt;X&gt;/I</tt>. The <em>Hilbert function</em> of <tt>K</tt>-algebra <tt>K&lt;X&gt;/I</tt> is a map from <tt>N</tt> to <tt>N</tt> defined by mapping <tt>i</tt> to <tt>dim(F_{i}/(F_{i} intersect I))-dim(F_{i-1}/(F_{i-1} intersect I))</tt>.
+
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>.
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
−
NCo.HF(Gb:LIST):LIST
+
NCo.HF(Gb:LIST[, DB:INT]):LIST
−
NCo.HF(Gb:LIST, DegreeBound: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 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 the function. The default coefficient field is <tt>Q</tt>. The default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.
+
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.
 
<itemize>
 
<itemize>
−
<item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K&lt;X&gt;</tt> which is a Groebner basis (w.r.t. a length compatible admissible ordering, say <tt>Ordering</tt>) of the two-sided ideal generated by Gb. Each polynomial is represented as a LIST of monomials, which are pairs 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 w.r.t. <tt>Ordering</tt>! In the case that Gb is a partical Groebner basis, the function enumerates pseudo values.</item>
+
<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>
−
<item>@param <em>DegreeBound:</em> (optional) a positive integer which is a degree bound of Hilbert funtion. <em>Note that</em> we set <tt>DegreeBound=32</tt> by default. Thus to compute all the values of the Hilbert function, it is necessary to set  <tt>DegreeBound</tt> to a larger enough number.</item>
+
 
−
<item>@return: a LIST of non-negative integers, which is a list of values of the Hilbert function of the K-algebra <tt>K&lt;X&gt;/(Gb)</tt>.</item>
+
<item>@return: a LIST of non-negative integers, which are values of the Hilbert function of the K-algebra <tt>K&lt;X&gt;/&lt;Gb&gt;</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 DB=32 by default. Thus, in the case that the <tt>K</tt>-dimension of <tt>K&lt;X&gt;/&lt;Gb&gt;</tt> is finite, it is necessary to set DB to a large enough INT in order to compute all the values of the Hilbert function.</item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>
Line 30: Line 33:
 
</description>
 
</description>
 
<seealso>
 
<seealso>
 +
<see>NCo.IsGB</see>
 +
<see>NCo.MB</see>
 
<see>NCo.SetFp</see>
 
<see>NCo.SetFp</see>
 
<see>NCo.SetOrdering</see>
 
<see>NCo.SetOrdering</see>
Line 37: Line 42:
 
<types>
 
<types>
 
<type>apcocoaserver</type>
 
<type>apcocoaserver</type>
 +
<type>ideal</type>
 
<type>groebner</type>
 
<type>groebner</type>
−
<type>ideal</type>
 
 
<type>non_commutative</type>
 
<type>non_commutative</type>
 
</types>
 
</types>

Revision as of 12:57, 30 April 2013

NCo.HF

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

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.

Syntax

NCo.HF(Gb: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 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(<quotes>xyzt</quotes>); 
NCo.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>]]]; 
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




Retrieved from "http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NCo.HF&oldid=13196"