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

From ApCoCoAWiki

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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/> | ||

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

<item>@param <em>BB</em> A border basis of an ideal.</item> | <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 BB.</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> | </itemize> | ||

<example> | <example> |

## Revision as of 15:52, 8 July 2009

## 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 := 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 Groebner Basis in CoCoA