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

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{{Version|1}}
 
<command>
 
<command>
    <title>BB.TransformGBIntoBB</title>
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  <title>BB.TransformGBIntoBB</title>
    <short_description>transform Groebner basis into border basis</short_description>
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  <short_description>Transforms a Groebner basis into a border basis.</short_description>
 +
 
 
<syntax>
 
<syntax>
 
BB.TransformGBIntoBB(GB:LIST of POLY):LIST of POLY
 
BB.TransformGBIntoBB(GB:LIST of POLY):LIST of POLY
 
</syntax>
 
</syntax>
    <description>
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  <description>
Let <tt>GB></tt> be a list of polynomials that form a <formula>\sigma</formula>-Groebner basis of a zero-dimensional ideal <formula>I</formula>. This function computes the <formula>\mathcal{O}_\sigma(I)</formula>-border basis of <formula>I</formula> by using the information provided by the given <formula>\sigma</formula>-Groebner basis.
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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.
 
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<par/>
 +
Let <tt>GB</tt> be a list of polynomials that form a <tt>sigma</tt>-Groebner basis of a zero-dimensional ideal <tt>I</tt>. This function computes the <tt>O_sigma(I)</tt>-border basis of <tt>I</tt> by using the information provided by the given <tt>sigma</tt>-Groebner basis.
 +
<itemize>
 +
  <item>@param <em>GB</em> A Groebner basis of a zero-dimensional ideal.</item>
 +
  <item>@return A list of polynomials that represents the border basis of the zero-dimensional ideal generated by the input polynomials in <tt>GB</tt>.</item>
 +
</itemize>
 
<example>
 
<example>
Use Z/(32003)[x,y,z],DegLex;
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Use ZZ/(32003)[x,y,z],DegLex;
 
I := Ideal(
 
I := Ideal(
 
4*x+5*y+6,
 
4*x+5*y+6,
Line 35: Line 42:
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
     </description>
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  </description>
     <see>BBasis</see>
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  <types>
    <see>GBasis</see>
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    <type>polynomial</type>
    <key>kaspar</key>
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    <type>groebner</type>
    <key>bb.transformgbintobb</key>
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     <type>borderbasis</type>
    <key>borderbasis.transformgbintobb</key>
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     <type>apcocoaserver</type>
    <wiki-category>Package_borderbasis</wiki-category>
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  </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>TransformGBIntoBB</key>
 +
  <key>BB.TransformGBIntoBB</key>
 +
  <key>borderbasis.TransformGBIntoBB</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.TransformGBIntoBB

Transforms a Groebner basis into a border basis.

Syntax

BB.TransformGBIntoBB(GB: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 GB be a list of polynomials that form a sigma-Groebner basis of a zero-dimensional ideal I. This function computes the O_sigma(I)-border basis of I by using the information provided by the given sigma-Groebner basis.

  • @param GB A Groebner basis of a zero-dimensional ideal.

  • @return A list of polynomials that represents the border basis of the zero-dimensional ideal generated by the input polynomials in GB.

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
);
GB := GBasis(I); -- compute a Groebner basis of I
BB := BB.TransformGBIntoBB(GB);
BB;

-------------------------------
[x + 8002y - 16000,
 xz + 8002yz - 16000z,
 xy + 8002y^2 - 16000y,
 y^2z - 5614yz^2 + 6179y^2 - 2246yz - 4492y - 3370z,
 y^3 + 12128yz^2 + 2045y^2 - 10508yz + 10240z^2 + 3337y - 8088z - 11495,
 xz^2 + 8002yz^2 - 16000z^2,
 xyz - 8984yz^2 + 277y^2 + 2809yz + 5615y - 11789z,
 xy^2 - 15160yz^2 + 5446y^2 + 13135yz - 12800z^2 - 12172y + 10110z + 6368,
 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,
 y^2z^2 + 1958yz^2 - 11982z^3 + 13714y^2 + 3833yz - 12303z^2 - 11335y + 4481z + 7925,
 xz^3 + 4083yz^2 - 14176z^3 - 8440y^2 - 10130yz + 10863z^2 - 14814y - 5151z - 9107,
 xyz^2 - 2446yz^2 - 1024z^3 - 1141y^2 - 12792yz + 7378z^2 + 6168y - 13602z + 14096]
-------------------------------


Introduction to CoCoAServer

Introduction to Groebner Basis in CoCoA

BB.BBasis

GBasis