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

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{{Version|1}}
 
<command>
 
<command>
 
<title>NC.IsGB</title>
 
<title>NC.IsGB</title>
 
<short_description>
 
<short_description>
−
Check whether a list of non-zero polynomials over a free monoid ring is a Groebner basis.
+
Check whether a LIST of non-zero polynomials is a Groebner basis in a non-commutative polynomial ring.  
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
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</syntax>
 
</syntax>
 
<description>
 
<description>
 +
Note that, given a word ordering, a set of non-zero polynomials <tt>G</tt> is called a <em>Groebner basis</em> of with respect to this ordering if the leading word set <tt>LW{G}</tt> generates the leading word ideal <tt>LW(&lt;G&gt;)</tt>. This function checks whether a given finite set of non-zero polynomial <tt>G</tt> is a Groebner basis by using the <tt>Buchberger Criterion</tt>, i.e. <tt>G</tt> is a Groebner basis if the S-polynomials of all obstructions of <tt>G</tt> have the zero normal remainder with respect to <tt>G</tt>.
 +
<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>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.
+
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 polynomials 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 non-commutative polynomials. 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>
−
<item>@return: a BOOL value which is True if <tt>G</tt> is a Groebner basis w.r.t. current ordering and False otherwise.</item>
+
<item>@return: a BOOL, which is True if <tt>G</tt> is a Groebner basis with respect to the current ordering and False otherwise.</item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>
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NC.SetX(<quotes>xyt</quotes>);
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Use ZZ/(2)[t,x,y];
−
F1 := [[1,<quotes>xx</quotes>], [-1,<quotes>yx</quotes>]]; 
+
G := [[[x^2], [y, x]], [[t, y], [x, y]], [[y, t], [x, y]], [[t, x], [x, t]],
−
F2 := [[1,<quotes>xy</quotes>], [-1,<quotes>ty</quotes>]]; 
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[[x, y, x], [y^2, x]], [[x, y^2], [y, x, y]], [[y, x, t], [y^2, x]]];
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F3 := [[1,<quotes>xt</quotes>], [-1, <quotes>tx</quotes>]]; 
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NC.SetOrdering("ELIM");  
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F4 := [[1,<quotes>yt</quotes>], [-1, <quotes>ty</quotes>]]; 
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NC.IsGB(G);
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G := [F1, F2,F3,F4];  
 
−
NC.IsGB(G); -- LLEX ordering (default ordering)
 
−
False
 
  
 +
True
 
-------------------------------
 
-------------------------------
−
NC.SetOrdering(<quotes>ELIM</quotes>);
+
NC.SetOrdering("LLEX");  
 
NC.IsGB(G);
 
NC.IsGB(G);
 +
 
False
 
False
−
 
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
 
</description>
 
</description>
 
<seealso>
 
<seealso>
−
<see>NC.Add</see>
+
<see>ApCoCoA-1:Use|Use</see>
−
<see>NC.Deg</see>
+
<see>ApCoCoA-1:NC.GB|NC.GB</see>
−
<see>NC.FindPolynomials</see>
+
<see>ApCoCoA-1:NC.LW|NC.LW</see>
−
<see>NC.GB</see>
+
<see>ApCoCoA-1:NC.RedGB|NC.RedGB</see>
−
<see>NC.HF</see>
+
<see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see>
−
<see>NC.Interreduction</see>
+
<see>ApCoCoA-1:NC.TruncatedGB|NC.TruncatedGB</see>
−
<see>NC.Intersection</see>
+
<see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
−
<see>NC.IsGB</see>
 
−
<see>NC.KernelOfHomomorphism</see>
 
−
<see>NC.LC</see>
 
−
<see>NC.LT</see>
 
−
<see>NC.LTIdeal</see>
 
−
<see>NC.MinimalPolynomial</see>
 
−
<see>NC.Multiply</see>
 
−
<see>NC.NR</see>
 
−
<see>NC.ReducedGB</see>
 
−
<see>NC.SetFp</see>
 
−
<see>NC.SetOrdering</see>
 
−
<see>NC.SetRelations</see>
 
−
<see>NC.SetRules</see>
 
−
<see>NC.SetX</see>
 
−
<see>NC.Subtract</see>
 
−
<see>NC.UnsetFp</see>
 
−
<see>NC.UnsetOrdering</see>
 
−
<see>NC.UnsetRelations</see>
 
−
<see>NC.UnsetRules</see>
 
−
<see>NC.UnsetX</see>
 
−
<see>Introduction to CoCoAServer</see>
 
 
</seealso>
 
</seealso>
 
<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>
−
<key>gbmr.IsGB</key>
+
<key>ncpoly.IsGB</key>
 
<key>NC.IsGB</key>
 
<key>NC.IsGB</key>
 
<key>IsGB</key>
 
<key>IsGB</key>
−
<wiki-category>Package_gbmr</wiki-category>
+
<wiki-category>ApCoCoA-1:Package_ncpoly</wiki-category>
 
</command>
 
</command>

Latest revision as of 13:34, 29 October 2020

This article is about a function from ApCoCoA-1.

NC.IsGB

Check whether a LIST of non-zero polynomials is a Groebner basis in a non-commutative polynomial ring.

Syntax

NC.IsGB(G:LIST):BOOL

Description

Note that, given a word ordering, a set of non-zero polynomials G is called a Groebner basis of with respect to this ordering if the leading word set LW{G} generates the leading word ideal LW(<G>). This function checks whether a given finite set of non-zero polynomial G is a Groebner basis by using the Buchberger Criterion, i.e. G is a Groebner basis if the S-polynomials of all obstructions of G have the zero normal remainder with respect to G.

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. 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 BOOL, which is True if G is a Groebner basis with respect to the current ordering and False otherwise.

Example

Use ZZ/(2)[t,x,y];
G := [[[x^2], [y, x]], [[t, y], [x, y]], [[y, t], [x, y]], [[t, x], [x, t]], 
[[x, y, x], [y^2, x]], [[x, y^2], [y, x, y]], [[y, x, t], [y^2, x]]];
NC.SetOrdering("ELIM"); 
NC.IsGB(G);

True
-------------------------------
NC.SetOrdering("LLEX"); 
NC.IsGB(G);

False
-------------------------------

See also

Use

NC.GB

NC.LW

NC.RedGB

NC.SetOrdering

NC.TruncatedGB

Introduction to CoCoAServer




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