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 StableBBasis5

Stable Border Basis of ideal of points
 Syntax
 ``` StableBBasis5(Pts: LIST, Toler: LIST): RECORD[SOI: LIST, BBasis: LIST, AlmostVanishing: LIST, StableBBasisFound: BOOL] StableBBasis5(Pts: LIST, Toler: LIST, Gamma: RAT): RECORD[SOI: LIST, BBasis: LIST, AlmostVanishing: LIST, StableBBasisFound: BOOL] ```

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

This function returns a record containing a stable order ideal of the ideal of points, and a list of almost vanishing polynomials. If the cardinality of the order ideal is equal to the number of points, it is in fact a quotient basis, and in this case a stable border basis founded on it is also returned. The boolean field StableBBasisFound is set to true if a stable border basis was found, otherwise false.

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). The function builds the stable order ideal stepwise by testing, from a numerical point of view, the linear dependence of a set of vectors.

There is a third, optional argument: it is a real non negative number Gamma which is used for scaling the threshold on the admissible perturbation of the points. A value of Gamma << 1 should be used. If no value is specified then by default Gamma = 0.1

For a full description of the algorithms we refer to the paper J.Abbott, C.Fassino, L.Torrente Stable Border Bases for Ideals of Points (to appear in JSC or arXiv:07062316).

 Example
 ``` Pts := [[0.1,-1],[1,1],[2,3]]; Toler := [0.1,0.1]; StableBBasis5(Pts, Toler); Record[ AlmostVanishing = [ (...) ], BBasis = [ -3602879701896397/288230376151711744y^2 + x - 32425917317067571/72057594037927936y - 154923827181545063/288230376151711744, xy - 140512308373959475/288230376151711744y^2 - 39631676720860365/72057594037927936y + 10808639105689191/288230376151711744, y^3 - 3y^2 - y + 3, xy^2 - 580063632005319885/288230376151711744y^2 - 32425917317067571/72057594037927936y + 421536925121878425/288230376151711744], SOI = [1, y, y^2], StableBBasisFound = True] ------------------------------- Toler := [0.6, 0.6]: StableBBasis5(Pts, Toler); Record[AlmostVanishing = [.....], SOI = [1, y], StableBBasisFound = False] ------------------------------- ```

 See Also