Difference between revisions of "ApCoCoA-1:BB.TransformBBIntoGB"

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(Short description update.)
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{{Version|1}}
 
<command>
 
<command>
    <title>BB.TransformBBIntoGB</title>
+
  <title>BB.TransformBBIntoGB</title>
    <short_description>Transform a border basis into a Groebner basis.</short_description>
+
  <short_description>Transforms a border basis into a Groebner basis.</short_description>
 +
 
 
<syntax>
 
<syntax>
 
BB.TransformBBIntoGB(BB:LIST of POLY):LIST of POLY
 
BB.TransformBBIntoGB(BB:LIST of POLY):LIST of POLY
 
</syntax>
 
</syntax>
    <description>
+
  <description>
{{ApCoCoAServer}}
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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.
 
+
<par/>
Let <tt>BB</tt> be a list of polynomials that form a <formula>\mathcal{O}_\sigma(I)</formula>-border basis of a zero-dimensional ideal <formula>I</formula>. This function extracts the reduced <formula>\sigma</formula>-Groebner basis contained in the <formula>\mathcal{O}_\sigma(I)</formula>-border basis <tt>BB</tt> and returns it as a list of polynomials.
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Let <tt>BB</tt> be a list of polynomials that form a <tt>O_sigma(I)</tt>-border basis of a zero-dimensional ideal <tt>I</tt>. This function extracts the reduced <tt>sigma</tt>-Groebner basis contained in the <tt>O_sigma(I)</tt>-border basis <tt>BB</tt> and returns it as a list of polynomials.
 
+
<itemize>
 +
  <item>@param <em>BB</em> A border basis of an ideal.</item>
 +
  <item>@return A list of polynomials that represents the reduced Groebner basis of the ideal generated by the input polynomials in <tt>BB</tt>.</item>
 +
</itemize>
 
<example>
 
<example>
Use Z/(32003)[x,y,z],DegLex;
+
Use ZZ/(32003)[x,y,z],DegLex;
 
I := Ideal(
 
I := Ideal(
 
4*x+5*y+6,
 
4*x+5*y+6,
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x^2*y+3*x*y*z+x*z^2+15*x^2+x*y+9*y*z+7
 
x^2*y+3*x*y*z+x*z^2+15*x^2+x*y+9*y*z+7
 
);
 
);
BB := BBasis(I); -- compute a border basis of I
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BB := BB.BBasis(I); -- compute a border basis of I
 
GB := BB.TransformBBIntoGB(BB);
 
GB := BB.TransformBBIntoGB(BB);
 
GB;
 
GB;
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-------------------------------
 
-------------------------------
 
</example>
 
</example>
     </description>
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  </description>
     <see>BBasis</see>
+
  <types>
    <see>GBasis</see>
+
    <type>polynomial</type>
    <key>kaspar</key>
+
    <type>borderbasis</type>
    <key>bb.transformbbintogb</key>
+
     <type>groebner</type>
    <key>borderbasis.transformbbintogb</key>
+
     <type>apcocoaserver</type>
    <wiki-category>Package_borderbasis</wiki-category>
+
  </types>
 +
 
 +
  <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
 +
  <see>ApCoCoA-1:Introduction to Groebner Basis in CoCoA|Introduction to Groebner Basis in CoCoA</see>
 +
  <see>ApCoCoA-1:BB.BBasis|BB.BBasis</see>
 +
  <see>ApCoCoA-1:GBasis|GBasis</see>
 +
 
 +
  <key>TransformBBIntoGB</key>
 +
  <key>BB.TransformBBIntoGB</key>
 +
  <key>borderbasis.TransformBBIntoGB</key>
 +
  <wiki-category>ApCoCoA-1:Package_borderbasis</wiki-category>
 
</command>
 
</command>

Latest revision as of 09:43, 7 October 2020

This article is about a function from ApCoCoA-1.

BB.TransformBBIntoGB

Transforms a border basis into a Groebner basis.

Syntax

BB.TransformBBIntoGB(BB:LIST of POLY):LIST of POLY

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.

Let BB be a list of polynomials that form a O_sigma(I)-border basis of a zero-dimensional ideal I. This function extracts the reduced sigma-Groebner basis contained in the O_sigma(I)-border basis BB and returns it as a list of polynomials.

  • @param BB A border basis of an ideal.

  • @return A list of polynomials that represents the reduced Groebner basis of the ideal generated by the input polynomials in BB.

Example

Use ZZ/(32003)[x,y,z],DegLex;
I := Ideal(
4*x+5*y+6,
2*x^2*z+4*y^2*z+4*y*z^2+3*x*y+25*y^2+7*x*z+2*y-3*z,
x^2*y+3*x*y*z+x*z^2+15*x^2+x*y+9*y*z+7
);
BB := BB.BBasis(I); -- compute a border basis of I
GB := BB.TransformBBIntoGB(BB);
GB;

-------------------------------
[x + 8002y - 16000, y^2z - 5614yz^2 + 6179y^2 - 2246yz - 4492y - 3370z,
 y^3 + 12128yz^2 + 2045y^2 - 10508yz + 10240z^2 + 3337y - 8088z - 11495,
 z^4 - 928yz^2 + 15802z^3 - 8546y^2 - 13286yz - 15491z^2 - 13314y + 5553z - 11227,
 yz^3 - 9667yz^2 + 11342z^3 + 6752y^2 + 8104yz - 15091z^2 - 950y - 15081z + 885]
-------------------------------


Introduction to CoCoAServer

Introduction to Groebner Basis in CoCoA

BB.BBasis

GBasis