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

 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]
-------------------------------
```