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

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(New page: <command> <title>NC.ReducedGB</title> <short_description> Enumerate a reduced (partial) Groebner basis of a finitely generated two-sided ideal in a non-commutative polynomial ring. <par/>...)
 
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<command>
 
<command>
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<title>NC.ReducedGB</title>
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<title>NC.RedGB</title>
 
<short_description>
 
<short_description>
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Enumerate a reduced (partial) Groebner basis of a finitely generated two-sided ideal in a non-commutative polynomial ring.  
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Enumerate reduced (partial) Groebner bases of finitely generated two-sided ideals in a non-commutative polynomial ring.
 
<par/>
 
<par/>
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Note that, given an ideal <tt>I</tt> and an admissible ordering <tt>Ordering</tt>, a set of non-zero polynomials <tt>Gb</tt> is called a <em>Groebner basis</em> of <tt>I</tt> w.r.t. <tt>Ordering</tt> if the leading term set <tt>LT{Gb}</tt> (w.r.t. <tt>Ordering</tt>) generates the leading term ideal <tt>LT(I)</tt> (w.r.t. <tt>Ordering</tt>). A Groebner basis w.r.t. <tt>Ordering</tt> is called a <em> reduced Groebner basis</em> w.r.t. <tt>Ordering</tt> if (1) it is interreduced and (2) all polynomials in it are monic. Also note that each ideal has a unique reduced Groebner basis w.r.t. <tt>Ordering</tt>. However, it is not necessarily finite.
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Given a word ordering and a two-sided ideal <tt>I</tt>, a set of non-zero polynomials <tt>Gb</tt> is called a <em>Groebner basis</em> of <tt>I</tt> if the leading word set <tt>LT{Gb}</tt> generates the leading word ideal <tt>LT(I)</tt>. A Groebner basis <tt>Gb</tt> of <tt>I</tt> is <em>reduced</em> if <tt>Gb</tt> is interreduced (see <ref>NC.Interreduction</ref>) and all polynomials in <tt>Gb</tt> are monic. Note that each ideal has a unique (not necessarily finite) reduced Groebner basis.
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
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NC.RedGB(G:LIST):LIST
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NC.RedGB(G:LIST[, Optimize:INT, OFlag:INT, DB:INT, LB:INT]):LIST
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NC.RedGB(G:LIST, DegreeBound:INT, LoopBound:INT, Flag: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/>
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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>NC.SetFp</ref>, <ref>NC.SetX</ref> and <ref>NC.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.
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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.
 
<itemize>
 
<itemize>
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<item>@param <em>G</em>: a LIST of non-zero polynomials generating a two-sided ideal in <tt>K&lt;X&gt;</tt>. 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>]].</item>
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<item>@param <em>G</em>: a LIST of non-zero commutative polynomials that generates a two-sided ideal. 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 [].</item>
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<item>@return: a LIST of interreduced non-zero non-commutative polynomials, which is a reduced Groebner basis of the two-sided ideal generated by G if (1) there exists a finite Groebner basis and (2) the enumerating procedure was not interrupted by the degree bound DB or the loop bound LB, and is a reduced partial Groebner basis otherwise.</item>
 
</itemize>
 
</itemize>
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About the optional parameters: for most cases we do not know whether or not there exists a finite Groebner basis. Thue, the function has three optional parameters to interrupt the enumerating procedure. Note that at the moment <em>all</em> of the following three optional parameters must be used at the same time.
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Optional parameters:
 
<itemize>
 
<itemize>
−
<item>@param <em>DegreeBound:</em> a positive integer which gives a degree bound during the enumerating procedure. When the degree of the normal remainder of some <tt>S-polynomial</tt> reaches DegreeBound, the function stops the loop and returns a partial Groebner basis.</item>
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<item>@param <em>Optimize:</em> an INT between 0 and 31. The value of Optimize indicates which criteria are used in the Buchberger procedure. For the time being, the function supports 5 criteria as follows.
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<item>@param <em>LoopBound:</em> a positive integer which gives a bound for the main loop of the enumerating procedure. When it runs through the main loop LoopBound times, the function stops and returns a partial Groebner basis.</item>
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  The multiply criterion: Optimize=1;
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<item>@param <em>Flag:</em> a non-negative integer which is a multi-switch for the output of ApCoCoAServer. If Flag=0, the server prints nothing on the screen. If Flag=1, the server prints basic information on the enumerating procedure, such as the number of main loop that has been proceeded, the number of elements in partial Groebner basis, the number of unselected obstructions; the total number of obstructions, the number of selected obstructions, and the number of unnecessary obstructions. If Flag=2, beside the information as FLAG=1, the server also displays explicitly the elements in paritial Groebner basis and current selected <tt>S-polynonial</tt>. Note that the initial idea of using Flag is to trace and debug the enumerating procedure.</item>
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  The leading word criterion: Optimize=2;
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<item>@return: a LIST of polynomials, which is a reduced Groebner basis (w.r.t. the current ordering) of the two-sided ideal generated by <tt>G</tt> if (1) there exists a finite Groebner basis and (2) the enumerating procedure doesn't terminate due to reaching <tt>DegreeBound</tt> or <tt>LoopBound</tt>, and is a reduced partial Groebner basis otherwise.</item>
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  The tail reduction criterion: Optimize=4;
 +
  The backward criterion: Optimize=8;
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  Discarding redundant generators: Optimize=16;
 +
  (Reference: M. Kreuzer and X. Xiu, Non-Commutative Gebauer-Moeller Criteria, 2013.)
 +
One can choose none, one, or more than one criterion in the function. For instance, Optimize=0 means no criterion is used, Optimize=1 means applying the multiply criterion, Optimize=3 (1+2) means applying the multiply and the leading word criteria, etc. By default, Optimize=31, i.e. the function applies all criteria.</item>
 +
 
 +
<item>@param <em>OFlag:</em> an INT that indicates which information will be output by the ApCoCoAServer during the Buchberger enumerating procedure. If OFlag=1, the ApCoCoAServer outputs basic information, such as the number of enumerating steps that has been proceeded, the number of elements in partial Groebner basis, the degree of current selected obstruction, the number of unselected obstructions, the total number of obstructions, the number of selected obstructions, and the number of unnecessary obstructions. If Flag=2, besides the information as FLAG=1, the ApCoCoAServer also displays explicitly the elements in paritial Groebner basis and the current selected S-polynonial. Otherwise, the ApCoCoAServer outputs nothing during the procedure.</item>
 +
 
 +
<item>@param <em>DB:</em> a positive INT, which gives a degree bound of S-polynomials (or obstructions) during the Buchberger enumerating procedure. When the degree bound is reached, the enumerating procedure will be interrupted and return a partial Groebner basis.</item>
 +
 
 +
<item>@param <em>LB:</em> a positive INT, which gives a loop bound of enumerating steps. When the loop bound is reached,the enumerating procedure will be interrupted and return a partial Groebner basis.</item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>
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<types>
 
<types>
 
<type>apcocoaserver</type>
 
<type>apcocoaserver</type>
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<type>ideal</type>
 
<type>groebner</type>
 
<type>groebner</type>
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<type>ideal</type>
 
 
<type>non_commutative</type>
 
<type>non_commutative</type>
 
</types>
 
</types>
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<key>ncpoly.ReducedGB</key>
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<key>ncpoly.RedGB</key>
 
<key>NC.RedGB</key>
 
<key>NC.RedGB</key>
 
<key>RedGB</key>
 
<key>RedGB</key>
 
<wiki-category>Package_ncpoly</wiki-category>
 
<wiki-category>Package_ncpoly</wiki-category>
 
</command>
 
</command>

Revision as of 19:24, 26 April 2013

NC.RedGB

Enumerate reduced (partial) Groebner bases of finitely generated two-sided ideals in a non-commutative polynomial ring.

Given a word ordering and a two-sided ideal I, a set of non-zero polynomials Gb is called a Groebner basis of I if the leading word set LT{Gb} generates the leading word ideal LT(I). A Groebner basis Gb of I is reduced if Gb is interreduced (see NC.Interreduction) and all polynomials in Gb are monic. Note that each ideal has a unique (not necessarily finite) reduced Groebner basis.

Syntax

NC.RedGB(G:LIST[, Optimize:INT, OFlag:INT, DB:INT, LB: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 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 commutative polynomials that generates a two-sided ideal. 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 [].

  • @return: a LIST of interreduced non-zero non-commutative polynomials, which is a reduced Groebner basis of the two-sided ideal generated by G if (1) there exists a finite Groebner basis and (2) the enumerating procedure was not interrupted by the degree bound DB or the loop bound LB, and is a reduced partial Groebner basis otherwise.

Optional parameters:

  • @param Optimize: an INT between 0 and 31. The value of Optimize indicates which criteria are used in the Buchberger procedure. For the time being, the function supports 5 criteria as follows. 

     The multiply criterion: Optimize=1;
 The leading word criterion: Optimize=2;
 The tail reduction criterion: Optimize=4;
 The backward criterion: Optimize=8;
 Discarding redundant generators: Optimize=16;
 (Reference: M. Kreuzer and X. Xiu, Non-Commutative Gebauer-Moeller Criteria, 2013.)

    One can choose none, one, or more than one criterion in the function. For instance, Optimize=0 means no criterion is used, Optimize=1 means applying the multiply criterion, Optimize=3 (1+2) means applying the multiply and the leading word criteria, etc. By default, Optimize=31, i.e. the function applies all criteria.

  • @param OFlag: an INT that indicates which information will be output by the ApCoCoAServer during the Buchberger enumerating procedure. If OFlag=1, the ApCoCoAServer outputs basic information, such as the number of enumerating steps that has been proceeded, the number of elements in partial Groebner basis, the degree of current selected obstruction, the number of unselected obstructions, the total number of obstructions, the number of selected obstructions, and the number of unnecessary obstructions. If Flag=2, besides the information as FLAG=1, the ApCoCoAServer also displays explicitly the elements in paritial Groebner basis and the current selected S-polynonial. Otherwise, the ApCoCoAServer outputs nothing during the procedure.

  • @param DB: a positive INT, which gives a degree bound of S-polynomials (or obstructions) during the Buchberger enumerating procedure. When the degree bound is reached, the enumerating procedure will be interrupted and return a partial Groebner basis.

  • @param LB: a positive INT, which gives a loop bound of enumerating steps. When the loop bound is reached,the enumerating procedure will be interrupted and return a partial Groebner basis.

Example

NC.SetX(<quotes>xyzt</quotes>); 
F1 := [[1,<quotes>xx</quotes>], [-1,<quotes>yx</quotes>]];
F2 := [[1,<quotes>xy</quotes>], [-1,<quotes>ty</quotes>]];
F3 := [[1,<quotes>xt</quotes>], [-1, <quotes>tx</quotes>]];
F4 := [[1,<quotes>yt</quotes>], [-1, <quotes>ty</quotes>]];
G := [F1, F2,F3,F4]; 
NC.ReducedGB(G); -- over Q (default field), LLEX ordering (default ordering)
[[[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>]]]
-------------------------------
NC.SetFp(); -- set default Fp=F2
NC.ReducedGB(G); -- over F2, LLEX ordering
[[[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>]]]
-------------------------------
NC.SetFp(3);
NC.ReducedGB(G); -- over F3, LLEX ordering
[[[1, <quotes>yt</quotes>], [2, <quotes>ty</quotes>]], [[1, <quotes>xt</quotes>], [2, <quotes>tx</quotes>]], [[1, <quotes>xy</quotes>], [2, <quotes>ty</quotes>]], [[1, <quotes>xx</quotes>], [2, <quotes>yx</quotes>]], 
[[1, <quotes>tyy</quotes>], [2, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [2, <quotes>tyx</quotes>]]]
-------------------------------
NC.SetX(<quotes>xyzt</quotes>); 
F1 := [[1,<quotes>xx</quotes>], [-1,<quotes>yx</quotes>]];   
F2 := [[1,<quotes>xy</quotes>], [-1,<quotes>ty</quotes>]];  
F3 := [[1,<quotes>xt</quotes>], [-1, <quotes>tx</quotes>]];  
F4 := [[1,<quotes>yt</quotes>], [-1, <quotes>ty</quotes>]];  
F:= [F1, F2,F3,F4]; 
G1:=NC.ReducedGB(F);
G1;
NC.IsGB(G1); -- G1 is a reduced Groebner basis
Fa:=[[1,<quotes>ytx</quotes>],[-1,<quotes></quotes>]]; 
Append(G1,Fa); -- add one element to G1
G1;
NC.IsGB(G1); -- G1 is not a Groebner basis any longer
G2:=NC.ReducedGB(G1); -- compute reduced Groebner basis G2 generated by G1
G2;
G3:=NC.IsGB(G2);
G3:=NC.ReducedGB(G2); -- compute reduced Groebner basis G3 generated by G2
G3;
NC.IsGB(G3);
G2=G3; -- of course
[[[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>]]]
-------------------------------
True
-------------------------------
[[[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>]], [[1, <quotes>ytx</quotes>], [-1, <quotes></quotes>]]]
-------------------------------
False
-------------------------------
[[[1, <quotes>y</quotes>], [-1, <quotes>t</quotes>]], [[1, <quotes>x</quotes>], [-1, <quotes>t</quotes>]], [[1, <quotes>ttt</quotes>], [-1, <quotes></quotes>]]]
-------------------------------
[[[1, <quotes>y</quotes>], [-1, <quotes>t</quotes>]], [[1, <quotes>x</quotes>], [-1, <quotes>t</quotes>]], [[1, <quotes>ttt</quotes>], [-1, <quotes></quotes>]]]
-------------------------------
True
-------------------------------
True
-------------------------------

See also

NC.GB

NC.IsGB

NC.SetOrdering

NC.TruncatedGB

Introduction to CoCoAServer




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