Claudia Fassino

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Let X be a set of points whose coordinates are known with limited accuracy; our aim is to give a characterization of the vanishing ideal I(X) independent of the data uncertainty. We present a method to compute a polynomial basis B of I(X) which exhibits structural stability, that is, if e X is any set of points differing only slightly from X, there exists a(More)
Given a finite set X of points and a tolerance ε representing the maximum error on the coordinates of each point, we address the problemof computing a simple polynomial f whose zero-locus Z(f ) ‘‘almost’’ contains the points of X. We propose a symbolic–numerical method that, starting from the knowledge of X and ε, determines a polynomial f whose degree is(More)
Given X, a set of points whose coordinates are perturbed by errors, we want to determine a “numerically stable” basis B of the vanishing ideal I(X), i.e. for any permitted perturbation X̃ of the original set of points X, only slight changes to the coefficients in B are needed to produce a basis B̃ for the perturbed vanishing ideal I(X̃). We use border bases(More)
As part of the current movement of extending the classical concepts of Computational Commutative Algebra to the “empirical” case (see [7]), we present a new numerical approach to characterizing the vanishing ideal associated to a set of limited precision points. A similar concept has already been presented in recent literature: for instance in [1], [2],(More)
Given a set X of “empirical” points, whose coordinates are perturbed by errors, we analyze whether it contains redundant information, that is whether some of its elements could be represented by a single equivalent point. If this is the case, the empirical information associated to X could be described by fewer points, chosen in a suitable way. We present(More)
From the numerical point of view, given a set X ⊂ Rn of s points whose coordinates are known with only limited precision, each set e X of s 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 ε on the data error, computes a set G of(More)
Given a finite set of points X ⊂ ℝ n $\mathbb {X}\subset \mathbb {R}^{n}$ , one may ask for polynomials p which belong to a subspace V and which attain given values at the points of X $\mathbb {X}$ . We focus on subspaces V of ℝ [ x 1 , … , x n ] $ \mathbb {R}[x_{1},\ldots ,x_{n}]$ , generated by low order monomials. Such V were computed by the(More)