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

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(New page: <command> <title>NCo.IsGB</title> <short_description> Check whether a finite list (set) of non-zero polynomials in a free monoid ring is a Groebner basis. <par/> Note that, given an ideal...)
 
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{{Version|1}}
 
<command>
 
<command>
 
<title>NCo.IsGB</title>
 
<title>NCo.IsGB</title>
 
<short_description>
 
<short_description>
Check whether a finite list (set) of non-zero polynomials in a free monoid ring is a Groebner basis.
+
Check whether a finite LIST of non-zero polynomials in a free monoid ring is a Groebner basis.  
<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>.
 
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
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</syntax>
 
</syntax>
 
<description>
 
<description>
 +
Note that, given an ideal <tt>I</tt> and a word ordering <tt>Ordering</tt>, a set of non-zero polynomials <tt>G</tt> is called a <em>Groebner basis</em> of <tt>I</tt> with respect to <tt>Ordering</tt> if the leading word set <tt>LW{G}</tt> generates the leading word ideal <tt>LW(I)</tt>. The function checks whether a given finite LIST of non-zero polynomials <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 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>NCo.SetFp</ref>, <ref>NCo.SetX</ref> and <ref>NCo.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 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>ApCoCoA-1:NCo.SetFp|NCo.SetFp</ref>, <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|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 ("LLEX"). For more information, please check the relevant 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>
+
<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 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,"xy"], [-1, "y"], [1,""]].</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 word ordering and False otherwise.</item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>
NCo.SetX(<quotes>xyt</quotes>);   
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NCo.SetX("xyt");   
F1 := [[1,<quotes>xx</quotes>], [-1,<quotes>yx</quotes>]];   
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F1 := [[1,"xx"], [-1,"yx"]];   
F2 := [[1,<quotes>xy</quotes>], [-1,<quotes>ty</quotes>]];   
+
F2 := [[1,"xy"], [-1,"ty"]];   
F3 := [[1,<quotes>xt</quotes>], [-1, <quotes>tx</quotes>]];   
+
F3 := [[1,"xt"], [-1, "tx"]];   
F4 := [[1,<quotes>yt</quotes>], [-1, <quotes>ty</quotes>]];   
+
F4 := [[1,"yt"], [-1, "ty"]];   
 
G := [F1, F2,F3,F4];  
 
G := [F1, F2,F3,F4];  
 
NCo.IsGB(G); -- LLEX ordering (default ordering)
 
NCo.IsGB(G); -- LLEX ordering (default ordering)
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False
 
False
 
-------------------------------
 
-------------------------------
NCo.SetOrdering(<quotes>ELIM</quotes>);
+
NCo.SetOrdering("ELIM");
 
NCo.IsGB(G);
 
NCo.IsGB(G);
  
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</description>
 
</description>
 
<seealso>
 
<seealso>
<see>NCo.GB</see>
+
<see>ApCoCoA-1:NCo.GB|NCo.GB</see>
<see>NCo.ReducedGB</see>
+
<see>ApCoCoA-1:NCo.LW|NCo.LW</see>
<see>NCo.SetFp</see>
+
<see>ApCoCoA-1:NCo.ReducedGB|NCo.ReducedGB</see>
<see>NCo.SetOrdering</see>
+
<see>ApCoCoA-1:NCo.SetFp|NCo.SetFp</see>
<see>NCo.SetX</see>
+
<see>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</see>
<see>NCo.TruncatedGB</see>
+
<see>ApCoCoA-1:NCo.SetX|NCo.SetX</see>
<see>Introduction to CoCoAServer</see>
+
<see>ApCoCoA-1:NCo.TruncatedGB|NCo.TruncatedGB</see>
 +
<see>ApCoCoA-1:Introduction to CoCoAServer|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>
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<key>NCo.IsGB</key>
 
<key>NCo.IsGB</key>
 
<key>IsGB</key>
 
<key>IsGB</key>
<wiki-category>Package_gbmr</wiki-category>
+
<wiki-category>ApCoCoA-1:Package_gbmr</wiki-category>
 
</command>
 
</command>

Latest revision as of 13:40, 29 October 2020

This article is about a function from ApCoCoA-1.

NCo.IsGB

Check whether a finite LIST of non-zero polynomials in a free monoid ring is a Groebner basis.

Syntax

NCo.IsGB(G:LIST):BOOL

Description

Note that, given an ideal I and a word ordering Ordering, a set of non-zero polynomials G is called a Groebner basis of I with respect to Ordering if the leading word set LW{G} generates the leading word ideal LW(I). The function checks whether a given finite LIST of non-zero polynomials G is a Groebner basis by using the Buchberger Criterion, i.e. G is a Groebner basis if the S-polynomials of all obstructions 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 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 G: a LIST of non-zero polynomials in K<X>. 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,""]].

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

Example

NCo.SetX("xyt");  
F1 := [[1,"xx"], [-1,"yx"]];   
F2 := [[1,"xy"], [-1,"ty"]];  
F3 := [[1,"xt"], [-1, "tx"]];  
F4 := [[1,"yt"], [-1, "ty"]];  
G := [F1, F2,F3,F4]; 
NCo.IsGB(G); -- LLEX ordering (default ordering)

False
-------------------------------
NCo.SetOrdering("ELIM");
NCo.IsGB(G);

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

See also

NCo.GB

NCo.LW

NCo.ReducedGB

NCo.SetFp

NCo.SetOrdering

NCo.SetX

NCo.TruncatedGB

Introduction to CoCoAServer