# ApCoCoA-1:Num.RatPoints

## Num.RatPoints

Computes the zero set of an exact zero dimensional border basis. The zeros are computed approximately using the eigenvalues of the transposed multiplication matrices.

### Syntax

```Num.RatPoints(BB:LIST, OrderIdeal:LIST)):LIST of MAT
```

### 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.

This command computes a set of points, which are the zeros of an exact border basis. This border basis is close to the approximate border basis AppBB. The set of (complex) points is represented as two matrices. The j-th column of the first matrix gives the real part of a point and the j-th column of the second matrix gives the imaginary part. For computation the function is using the Num.EigenValues command.

• @param AppBB An approximate border basis.

• @param OrderIdeal The associated order ideal

• @return A set of points in matrix form described above.

#### Example

```Use P::=QQ[x,y,z];

Points := Mat([[2/3,0,0],[0,10,0],[0,0,1/3]]);
R:=Num.ABM(Points, 0);
Dec(Num.RatPoints(R[1],R[2]),2);

-- CoCoAServer: computing Cpu Time = 0
-------------------------------
-- CoCoAServer: computing Cpu Time = 0.016
-------------------------------
[Mat([
[<quotes>0.66</quotes>, <quotes>0.00</quotes>, <quotes>0</quotes>],
[<quotes>0</quotes>, <quotes>0</quotes>, <quotes>10</quotes>],
[<quotes>0</quotes>, <quotes>0.33</quotes>, <quotes>0</quotes>]
]), Mat([
[<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0</quotes>],
[<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0</quotes>],
[<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0</quotes>]
])]
-------------------------------

```