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

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(New page: <command> <title>NCo.MB</title> <short_description> Enumerate Macaulay basis of a <tt>K</tt>-algebra. </short_description> <syntax> NCo.MB(Gb:LIST):LIST NCo.MB(Gb:LIST, DegreeBound:INT):LI...)
 
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<title>NCo.MB</title>
 
<title>NCo.MB</title>
 
<short_description>
 
<short_description>
−
Enumerate Macaulay basis of a <tt>K</tt>-algebra.
+
Enumerate a Macaulay's basis of a finitely generated <tt>K</tt>-algebra.
 +
<par/>
 +
Given a two-sided ideal <tt>I</tt> in a free monoid ring <tt>K&lt;X&gt;</tt>, we can consider the <tt>K</tt>-algebra <tt>K&lt;X&gt;/I</tt> as a <tt>K</tt>-vector space. Moreover, let <tt>G</tt> be a Groebner basis of <tt>I</tt>, and let <tt>B</tt> be the set of all words which are not a multiple of any word in the leading word set <tt>LW{G}</tt>. Then the residue class of the words in <tt>B</tt> form a <tt>K</tt>-basis, called <em>Macaulay's basis</em>, of <tt>K&lt;X&gt;/I</tt>. For the sake of computing the values of the Hilbert function (see <ref>NCo.HF</ref>) of <tt>K&lt;X&gt;/I</tt>, in this function we require that <tt>G</tt> has to be a Groebner basis with respect to a length compatible word ordering (see <ref>NCo.SetOrdering</ref>).
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
−
NCo.MB(Gb:LIST):LIST
+
NCo.MB(Gb:LIST[, DB:INT]):LIST
−
NCo.MB(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. Default coefficient field is <tt>Q</tt>. 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 using 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 <tt>length compatible</tt> 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 a pseudo basis.</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 a pseudo basis.</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 the whole Macaulay basis, it is necessary to set  <tt>DegreeBound</tt> to a larger enough number.</item>
+
 
−
<item>@return: a LIST of terms, which forms the Macaulay basis of the K-algebra <tt>K&lt;X&gt;/(Gb)</tt> w.r.t. <tt>Ordering</tt>.</item>
+
<item>@return: a LIST of words forming a Macaulay's basis 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 lengths of words. Note that we set DB=32 by default. Thus, in the case that <tt>K&lt;X&gt;/&lt;Gb&gt;</tt> has a finite Macaulay's basis, it is necessary to set DB to a large enough INT in order to compute the whole Macaulay's basis.</item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>
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</description>
 
</description>
 
<seealso>
 
<seealso>
 +
<see>NCo.HF</see>
 +
<see>NCo.IsGB</see>
 
<see>NCo.SetFp</see>
 
<see>NCo.SetFp</see>
 
<see>NCo.SetOrdering</see>
 
<see>NCo.SetOrdering</see>
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<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 12:34, 30 April 2013

NCo.MB

Enumerate a Macaulay's basis of a finitely generated K-algebra.

Given a two-sided ideal I in a free monoid ring K<X>, we can consider the K-algebra K<X>/I as a K-vector space. Moreover, let G be a Groebner basis of I, and let B be the set of all words which are not a multiple of any word in the leading word set LW{G}. Then the residue class of the words in B form a K-basis, called Macaulay's basis, of K<X>/I. For the sake of computing the values of the Hilbert function (see NCo.HF) of K<X>/I, in this function we require that G has to be a Groebner basis with respect to a length compatible word ordering (see NCo.SetOrdering).

Syntax

NCo.MB(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 using 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 a pseudo basis.

  • @return: a LIST of words forming a Macaulay's basis of the K-algebra K<X>/<Gb>.

Optional parameter:

  • @param DB: a positive INT which is a degree bound of the lengths of words. Note that we set DB=32 by default. Thus, in the case that K<X>/<Gb> has a finite Macaulay's basis, it is necessary to set DB to a large enough INT in order to compute the whole Macaulay's basis.

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.MB(Gb,3);
[[<quotes></quotes>], [<quotes>t</quotes>, <quotes>z</quotes>, <quotes>y</quotes>, <quotes>x</quotes>], [<quotes>tt</quotes>, <quotes>tz</quotes>, <quotes>ty</quotes>, <quotes>tx</quotes>, <quotes>zt</quotes>, <quotes>zz</quotes>, <quotes>zy</quotes>, <quotes>zx</quotes>, <quotes>yz</quotes>, <quotes>yy</quotes>, <quotes>yx</quotes>, <quotes>xz</quotes>], 
[<quotes>ttt</quotes>, <quotes>ttz</quotes>, <quotes>tty</quotes>, <quotes>ttx</quotes>, <quotes>tzt</quotes>, <quotes>tzz</quotes>, <quotes>tzy</quotes>, <quotes>tzx</quotes>, <quotes>tyz</quotes>, <quotes>tyx</quotes>, <quotes>txz</quotes>, <quotes>ztt</quotes>, <quotes>ztz</quotes>, <quotes>zty</quotes>, <quotes>ztx</quotes>,
<quotes>zzt</quotes>, <quotes>zzz</quotes>, <quotes>zzy</quotes>, <quotes>zzx</quotes>, <quotes>zyz</quotes>, <quotes>zyy</quotes>, <quotes>zyx</quotes>, <quotes>zxz</quotes>, <quotes>yzt</quotes>, <quotes>yzz</quotes>, <quotes>yzy</quotes>, <quotes>yzx</quotes>, <quotes>yyz</quotes>, <quotes>yyy</quotes>,
 <quotes>yxz</quotes>, <quotes>xzt</quotes>, <quotes>xzz</quotes>, <quotes>xzy</quotes>, <quotes>xzx</quotes>]]
-------------------------------

See also

NCo.HF

NCo.IsGB

NCo.SetFp

NCo.SetOrdering

NCo.SetX

Introduction to CoCoAServer




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