Difference between revisions of "ApCoCoA-1:CharP.GBasisModSquares"

From ApCoCoAWiki
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   <command>
 
   <command>
 
     <title>Char2.GBasisModSquares</title>
 
     <title>Char2.GBasisModSquares</title>
     <short_description>Computing a Groebner Basis of a given ideal intersected with x^2-x for all indeterminates x.</short_description>
+
     <short_description>Computing a Groebner Basis of a given ideal intersected with <tt>x^2-x</tt> for all indeterminates <tt>x</tt>.</short_description>
 
<syntax>
 
<syntax>
 
Char2.GBasisModSquares(Ideal:IDEAL):LIST
 
Char2.GBasisModSquares(Ideal:IDEAL):LIST
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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/>
This function returns the reduced Groebner basis for the given ideal intersected with the ideal generated by x^2-x for all indeterminates. If x^2-x for  
+
This function returns the reduced Groebner basis for the given ideal intersected with the ideal generated by <tt>x^2-x</tt> for all indeterminates. If <tt>x^2-x</tt> for all indeterminates is in the ideal (e.g. the set of zeros is a subset of <tt>{0,1}^n</tt>) this method should produce the Groebner Basis much faster!
all indeterminates is in the ideal (e.g. the set of zeros is a subset of {0,1}^n) this method should produce the Groebner Basis much faster!
+
Please be aware, that this is much more efficient if the term ordering is Lex, DegLex or DegRevLex. Otherwise, first a DegRevLex Groebner Basis is computed and then transformed with the <ref>FGLM.FGLM</ref>-algorithm.  
Please be aware, that this is much more efficient if the term ordering is Lex, DegLex or DegRevLex. Otherwise, first a DegRevLex Groebner Basis is computed and then transformed with the FGLM-algorithm.  
 
  
 
<itemize>
 
<itemize>
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<item>@return The reduced Groebner Basis of the given ideal.</item>
 
<item>@return The reduced Groebner Basis of the given ideal.</item>
 
</itemize>
 
</itemize>
 +
 +
<example>
 +
Use R::=QQ[x,y,z];
 +
I:=Ideal(x-y^2,x^2+xy,y^3);
 +
GBasis(I);
 +
 +
[x^2 + xy, -y^2 + x, -xy]
 +
-------------------------------
 +
Use Z::=ZZ[x,y,z];
 +
-- WARNING: Coeffs are not in a field
 +
-- GBasis-related computations could fail to terminate or be wrong
 +
 +
-------------------------------
 +
I:=Ideal(x-y^2,x^2+xy,y^3);
 +
Char2.GBasisModSquares(I);
 +
-- WARNING: Coeffs are not in a field
 +
-- GBasis-related computations could fail to terminate or be wrong
 +
-- CoCoAServer: computing Cpu Time = 0
 +
-------------------------------
 +
[y, x]
 +
-------------------------------
 +
</example>
 +
 
     </description>
 
     </description>
  

Revision as of 07:55, 14 July 2009

Char2.GBasisModSquares

Computing a Groebner Basis of a given ideal intersected with x^2-x for all indeterminates x.

Syntax

Char2.GBasisModSquares(Ideal:IDEAL):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.

This function returns the reduced Groebner basis for the given ideal intersected with the ideal generated by x^2-x for all indeterminates. If x^2-x for all indeterminates is in the ideal (e.g. the set of zeros is a subset of {0,1}^n) this method should produce the Groebner Basis much faster!

Please be aware, that this is much more efficient if the term ordering is Lex, DegLex or DegRevLex. Otherwise, first a DegRevLex Groebner Basis is computed and then transformed with the FGLM.FGLM-algorithm.

  • @param Ideal An Ideal.

  • @return The reduced Groebner Basis of the given ideal.

Example

Use R::=QQ[x,y,z];
I:=Ideal(x-y^2,x^2+xy,y^3);
GBasis(I);

[x^2 + xy, -y^2 + x, -xy]
-------------------------------
Use Z::=ZZ[x,y,z];
-- WARNING: Coeffs are not in a field
-- GBasis-related computations could fail to terminate or be wrong

-------------------------------
I:=Ideal(x-y^2,x^2+xy,y^3);
Char2.GBasisModSquares(I);
-- WARNING: Coeffs are not in a field
-- GBasis-related computations could fail to terminate or be wrong
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
[y, x]
-------------------------------


See also

FGLM.FGLM

GBasis

Introduction to CoCoAServer

Introduction to Groebner Basis in CoCoA

Representation of finite fields