Almost Vanishing Polynomials for Sets of Limited Precision Points

Details Almost Vanishing Polynomials for Sets of Limited Precision Points Almost Vanishing Polynomials for Sets of Limited Precision Points

2008-07-22

arxiv.org/abs/0807.3412v2

Journal of Symbolic Computation, 45, 2010, 19--37

Mathematics

Commutative Algebra

21 pages -- It is an improvement of the paper An approximation of the Groebner basis of ideals of perturbed points, part I --

Scientific paper

Let X be a set of s points whose coordinates are known with only limited From the numerical point of view, given a set X of s real points whose coordinates are known with only limited precision, each set X* of real points whose elements differ from those of X of a quantity less than the data uncertainty can be considered equivalent to X. We present an algorithm that, given X and a tolerance Tol on the data error, computes a set G of polynomials such that each element of G "almost vanishing" at X and at all its equivalent sets X*. Even if G is not, in the general case, a basis of the vanishing ideal I(X), we show that, differently from the basis of I(X) that can be greatly influenced by the data uncertainty, G can determine a geometrical configuration simultaneously characterizing the set X and all its equivalent sets X*.

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