# ApCoCoA-1:Num.ABM

## Num.BBasisOfPoints

Border basis of almost vanishing ideal for a set of points.

### Syntax

```Num.BBasisOfPoints(Points:MAT, Tau:RAT, GetO:BOOL):Object
Num.BBasisOfPoints(Points:MAT, Tau:RAT, GetO:BOOL, Delta:RAT,  NormalizeType:INT, RREFNormalizeType:INT, RREFUseEps:BOOL, RREFType:INT):Object
```

### 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.

• @param Points The points for which a border basis is computed.

• @param Tau A positive rational number describing which singular values should be treated as 0 (smaller values for tau lead to bigger errors of the polynomials evaluated at the point set). Tau should be in the interval (0,1). As a rule of thumb, Tau is the expected percentage of error on the input points.

• @param GetO A Boolean to choose the output. If it is true, the command returns a list of two values: the first contains the border basis, the second one a vector space basis of P/I comprising those power products lying outside the leading term ideal of I. If GetO is false, the function returns only the border basis (not in a list).

• @return The return value depends on the parameter GetO: GetO=FALSE: The border basis of the given points as a list of polynomials. GetO=TRUE: A list of two results. First the border basis as a list of polynomials, second the vector space basis of P/I as a list of terms.

The following parameters are optional:

• @param Delta A positiv rational number. Delta describes the computing precision. In different steps, it is crucial, if a value is 0 or not. The algorithm assumes every value in [-Delta, Delta] to be 0. The default value for Delta is 0.00000000001.

• @param NormalizeType A integer of the range 1..4. The default value is 1. This parameter describes, if / how the input points are normalized. If NormalizeType equals 1, each coordinate is divided by the maximal absolut value of the matrix's corresponding column. This ensures that all point's coordinates are in [-1,1]. With NormalizeType=2 no normalization is done at all. NormalizeType=3 shifts each coordinate to [-1,1]. So it's minimum is mapped to -1 and the maximum to one, describing a unique affine mapping. The last option is NormalizeType=4. In this case, each coordinate is normalized, using the column's euclidian norm. Due to backward compatibility, the default is 1, although 3 is in most cases a better choice.

• @param RREFNormalizeType Describes, how in each RREF steps the columns are normalized. The options correspond to the ones for NormalizeType and the default is 1 again.

• @param RREFUseEps must be either true or false! If RREFUseEps is true, the given Delta is used within the RREF to decide if a value equals 0 or not. If this parameter is false, a replacement value for Delta is used, which is based on the matrix's norm.

• @param RREFType This must be 1 or 2. If RREFType=1, the rref operates column-wise. Otherwise it works row-wise. The default is 1.

This command computes a border basis of an almost vanishing ideal for a set of points.

The current ring has to be a ring over the rationals with a standard-degree

compatible term-ordering. The matrix Points contains the points: each

point is a row in the matrix, so the number of columns must equal the number of indeterminates in the current ring.

#### Example

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

Points := Mat([[1,0,0],[0,0,1],[0,2,0]]);
Num.BBasisOfPoints(Points,0.001,True);
-------------------------------
[[x + 9007199254740991/18014398509481984y + z - 1, z^2 - 9007199254740991/9007199254740992z, 1/2yz, xz, 1/4y^2 - 9007199254740991/18014398509481984y, 1/2xy], [y, z, 1]]
```