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Numerical Border Basis of ideal of points

 StableBBasisNBM5(Pts: LIST, Toler: LIST): RECORD[SOI: LIST, BBasis: LIST, AlmostVanishing: LIST, StableBBasisFound: BOOL]

This function is implemented in CoCoALib (i.e. it requires an active CoCoAServer).

This function returns a record containing a factor-closed set of power-products QB and a list of almost vanishing polynomials. If the cardinality of QB is equal to the number of points, it is in fact a quotient basis of the ideal of points, and in this case a border basis founded on it is also returned.

The first argument is a list of points in k-dimensional space, and the second argument is list of k positive tolerances (one for each dimension).

For a full description of the algorithms we refer to the paper C.Fassino Almost Vanishing Polynomials for Sets of Limited Precision Points (arXiv:0807.3412).

  Eps := [0.1, 0.1];
  Points := [[10, 0], [-10, 0], [0, 10], [0, -10], [7, 7], [-7, -7]];
  StableBBasisNBM5(Points, Eps);
Record[AlmostVanishing = [x^2 + 2/49xy + y^2 - 100, xy^2 + 49/51y^3
                         - 4900/51y, y^4 + 51xy - 100y^2],
       BBasis = [x^2 + 2/49xy + y^2 - 100,   xy^2 + 49/51y^3 - 4900/51y,
                 x^2y + 49/51y^3 - 4900/51y,
                 y^4 + 51xy - 100y^2, xy^3 - 49xy],
       QB = [1, y, x, y^2, xy, y^3],
       StableBBasisFound = True]

See Also