ApCoCoA-1:CharP.GBasisModSquares: Difference between revisions
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<title> | <command> | ||
<short_description>Computing a Groebner Basis of a given ideal intersected with x^2-x for all indeterminates x.</short_description> | <title>CharP.GBasisModSquares</title> | ||
<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> | ||
CharP.GBasisModSquares(Ideal:IDEAL):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/> | ||
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 <tt>Lex</tt>, <tt>DegLex</tt> or <tt>DegRevLex</tt>. Otherwise, first a DegRevLex Groebner Basis is computed and then transformed with the <ref>ApCoCoA-1:FGLM.FGLM|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); | |||
CharP.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> | ||
<seealso> | <seealso> | ||
<see>FGLM.FGLM</see> | <see>ApCoCoA-1:FGLM.FGLM|FGLM.FGLM</see> | ||
<see>GBasis</see> | <see>ApCoCoA-1:GBasis|GBasis</see> | ||
<see>Introduction to Groebner Basis in CoCoA</see> | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> | ||
<see>Representation of finite fields</see> | <see>ApCoCoA-1:Introduction to Groebner Basis in CoCoA|Introduction to Groebner Basis in CoCoA</see> | ||
<see>ApCoCoA-1:Representation of finite fields|Representation of finite fields</see> | |||
</seealso> | </seealso> | ||
<types> | <types> | ||
<type> | <type>apcocoaserver</type> | ||
<type>ideal</type> | <type>ideal</type> | ||
<type>groebner</type> | <type>groebner</type> | ||
| Line 32: | Line 56: | ||
<key>gbasismodsquares</key> | <key>gbasismodsquares</key> | ||
<key> | <key>charP.gbasismodsquares</key> | ||
<wiki-category> | <key>finite field</key> | ||
<wiki-category>ApCoCoA-1:Package_charP</wiki-category> | |||
</command> | </command> | ||
Latest revision as of 09:55, 7 October 2020
| This article is about a function from ApCoCoA-1. |
CharP.GBasisModSquares
Computing a Groebner Basis of a given ideal intersected with x^2-x for all indeterminates x.
Syntax
CharP.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); CharP.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
Introduction to Groebner Basis in CoCoA
Representation of finite fields
