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

From ApCoCoAWiki
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Check whether a LIST of non-zero polynomials is a Groebner basis in a non-commutative polynomial ring.  
 
Check whether a LIST of non-zero polynomials is a Groebner basis in a non-commutative polynomial ring.  
 
<par/>
 
<par/>
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>). The function check 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 all the <tt>S-polynomials</tt> of obstructions have the zero normal remainder w.r.t. <tt>G</tt>.
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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>LT{G}</tt> generates the leading word ideal <tt>LT(&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 <tt>S-polynomials</tt> of all obstructions of <tt>G</tt> have the zero normal remainder with respect to <tt>G</tt>.
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
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<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.
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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>
<item>@param <em>G</em>: a LIST of non-zero 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. the current ordering and False otherwise.</item>
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<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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</description>
 
</description>
 
<seealso>
 
<seealso>
 +
<see>Use</see>
 
<see>NC.GB</see>
 
<see>NC.GB</see>
 
<see>NC.ReducedGB</see>
 
<see>NC.ReducedGB</see>

Revision as of 12:43, 26 April 2013

NC.IsGB

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

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 LT{G} generates the leading word ideal LT(<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.

Syntax

NC.IsGB(G:LIST):BOOL

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 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

NC.SetX(<quotes>xyt</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.IsGB(G); -- LLEX ordering (default ordering)

False
-------------------------------
NC.SetOrdering(<quotes>ELIM</quotes>);
NC.IsGB(G);

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

See also

Use

NC.GB

NC.ReducedGB

NC.SetOrdering

NC.TruncatedGB

Introduction to CoCoAServer