# ApCoCoA-1:BB.TransformGBIntoBB

## BB.TransformGBIntoBB

Transform 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 <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.

#### Example

Use Z/(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]
-------------------------------