# ApCoCoA-1:Num.ABM

## Numerical.BBasisOfPoints

Border basis of almost vanishing ideal for a set of points

### Syntax

$numerical.BBasisOfPoints(Points, Epsilon, GetO):Object

### Description

This command computes a border basis of an almost vanishing ideal for a set of points using the algorithm described in the paper

D. Heldt, M. Kreuzer, H. Poulisse, S.Pokutta:Approximate Computation of Zero-Dimensional IdealsSubmitted: August 2006

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. Epsilon is a rational >0 describing which singular values should be treated as 0 (smaller values for Epsilon lead to bigger errors of the polynomials evaluated at the point set). Epsilon should be in the interval (0,1). As a rule of thumb, Epsilon is the expected percentage of error on the input points. GetO must be either True or False. 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).

#### Example

Points := Mat([[1,0,0],[0,0,1],[0,2,0]]); $numerical.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]]

### See also

Numerical.GBasisOfPointsInIdeal

Numerical.BBasisOfPointsInIdeal

Numerical.HBasisOfPointsInIdeal

Numerical.FirstVanishingRelations

Numerical.FirstVanishingRelationsInIdeal