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

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{{Version|1}}
 
<command>
 
<command>
 
<title>NC.MB</title>
 
<title>NC.MB</title>
 
<short_description>
 
<short_description>
 
Enumerate a Macaulay's basis of a finitely generated <tt>K</tt>-algebra.
 
Enumerate a Macaulay's basis of a finitely generated <tt>K</tt>-algebra.
<par/>
 
Given a finitely generated two-sided ideal <tt>I</tt> in a finitely generated non-commutative polynomial ring <tt>P</tt> over <tt>K</tt>, we can consider the <tt>K</tt>-algebra <tt>P/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>LT{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>P/I</tt>. For the sake of computing the values of the Hilbert function (see <ref>NC.HF</ref>) of <tt>P/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>NC.SetOrdering</ref>).
 
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
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</syntax>
 
</syntax>
 
<description>
 
<description>
 +
Given a two-sided ideal <tt>I</tt> in a non-commutative polynomial ring <tt>P</tt> over <tt>K</tt>, we can consider the <tt>K</tt>-algebra <tt>P/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 a <em>Macaulay's basis</em>, of <tt>P/I</tt>. For the sake of computing the values of the Hilbert-Dehn function (see <ref>ApCoCoA-1:NC.HF|NC.HF</ref>) of <tt>P/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>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</ref>).
 +
<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 non-commutative polynomial ring (via the command <ref>Use</ref>) and word ordering (via the function <ref>NC.SetOrdering</ref>) before calling this function. The default word ordering is the length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant commands and functions.
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Please set non-commutative polynomial ring (via the command <ref>ApCoCoA-1:Use|Use</ref>) and word ordering (via the function <ref>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</ref>) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions.
 
<itemize>
 
<itemize>
 
<item>@param <em>G:</em> a LIST of non-zero non-commutative polynomials, which form a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial <tt>f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5</tt> is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial <tt>0</tt> is represented as the empty LIST []. <em>Warning:</em> users should take responsibility to ensure that G is indeed a Groebner basis with respect to a length compatible word ordering!</item>
 
<item>@param <em>G:</em> a LIST of non-zero non-commutative polynomials, which form a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial <tt>f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5</tt> is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial <tt>0</tt> is represented as the empty LIST []. <em>Warning:</em> users should take responsibility to ensure that G is indeed a Groebner basis with respect to a length compatible word ordering!</item>
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Optional parameter:
 
Optional parameter:
 
<itemize>
 
<itemize>
<item>@param <em>DB:</em> a positive INT, which is a degree bound of the lengths of words. <em>Note that</em> we set <tt>DB=32</tt> by default. Thus, in the case that <tt>P/&lt;G&gt;</tt> has a finite Macaulay's basis, it is necessary to set <tt>DB</tt> to a large enough INT in order to compute the whole Macaulay's basis.</item>
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<item>@param <em>DB:</em> a positive INT, which is a degree bound of the lengths of words. <em>Note that</em> we set DB=32 by default. Thus, in the case that <tt>P/&lt;G&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>
NC.SetX(<quotes>xyzt</quotes>);
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Use ZZ/(2)[t,x,y];
NC.SetOrdering(<quotes>LLEX</quotes>);
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NC.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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F1 := [[x^2], [y,x]]; -- x^2+yx
[[1, <quotes>tyy</quotes>], [-1, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [-1, <quotes>tyx</quotes>]]];
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F2 := [[x,y], [t,y]]; -- xy+ty
 +
F3 := [[x,t], [t,x]]; -- xt+tx
 +
F4 := [[y,t], [t,y]]; -- yt+ty
 +
G := [F1, F2,F3,F4];
 +
Gb:=NC.GB(G);
 
NC.MB(Gb,3);
 
NC.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>],  
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[<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>,
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[[[1]], [[y], [x], [t]], [[y, t], [y, x], [y^2], [x, t], [t^2]],  
<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>,
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[[y^3], [y^2, x], [y^2, t], [x, t^2], [t^3]]]
<quotes>yxz</quotes>, <quotes>xzt</quotes>, <quotes>xzz</quotes>, <quotes>xzy</quotes>, <quotes>xzx</quotes>]]
 
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
 
</description>
 
</description>
 
<seealso>
 
<seealso>
<see>Use</see>
+
<see>ApCoCoA-1:Use|Use</see>
<see>NC.HF</see>
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<see>ApCoCoA-1:NC.HF|NC.HF</see>
<see>NC.IsGB</see>
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<see>ApCoCoA-1:NC.IsGB|NC.IsGB</see>
<see>NC.SetOrdering</see>
+
<see>ApCoCoA-1:NC.LW|NC.LW</see>
<see>Introduction to CoCoAServer</see>
+
<see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see>
 +
<see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
 
</seealso>
 
</seealso>
 
<types>
 
<types>
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<key>NC.MB</key>
 
<key>NC.MB</key>
 
<key>MB</key>
 
<key>MB</key>
<wiki-category>Package_ncpoly</wiki-category>
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<wiki-category>ApCoCoA-1:Package_ncpoly</wiki-category>
 
</command>
 
</command>

Latest revision as of 13:35, 29 October 2020

This article is about a function from ApCoCoA-1.

NC.MB

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

Syntax

NC.MB(G:LIST[, DB:INT]):LIST

Description

Given a two-sided ideal I in a non-commutative polynomial ring P over K, we can consider the K-algebra P/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 a Macaulay's basis, of P/I. For the sake of computing the values of the Hilbert-Dehn function (see NC.HF) of P/I, in this function we require that G has to be a Groebner basis with respect to a length compatible word ordering (see NC.SetOrdering).

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 non-commutative polynomial ring (via the command Use) and word ordering (via the function NC.SetOrdering) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions.

  • @param G: a LIST of non-zero non-commutative polynomials, which form a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5 is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial 0 is represented as the empty LIST []. Warning: users should take responsibility to ensure that G is indeed a Groebner basis with respect to a length compatible word ordering!

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

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 P/<G> 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

Use ZZ/(2)[t,x,y];
NC.SetOrdering("LLEX"); 
F1 := [[x^2], [y,x]]; -- x^2+yx
F2 := [[x,y], [t,y]]; -- xy+ty
F3 := [[x,t], [t,x]]; -- xt+tx
F4 := [[y,t], [t,y]]; -- yt+ty
G := [F1, F2,F3,F4]; 
Gb:=NC.GB(G);
NC.MB(Gb,3);

[[[1]], [[y], [x], [t]], [[y, t], [y, x], [y^2], [x, t], [t^2]], 
[[y^3], [y^2, x], [y^2, t], [x, t^2], [t^3]]]
-------------------------------

See also

Use

NC.HF

NC.IsGB

NC.LW

NC.SetOrdering

Introduction to CoCoAServer