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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 $widetilde X$ is any set of points differing only slightly from $X$, there exists a polynomial set $widetilde B$ structurally similar to $B$, which is a basis of the perturbed ideal $ I(widetilde X)$.
In this paper we consider the problem of computing all possible order ideals and also sets connected to 1, and the corresponding border bases, for the vanishing ideal of a given finite set of points. In this context two different approaches are discu
We present an algorithm for computing Groebner bases of vanishing ideals of points that is optimized for the case when the number of points in the associated variety is less than the number of indeterminates. The algorithm first identifies a set of e
In modeling physical systems it is sometimes useful to construct border bases of 0-dimensional polynomial ideals which are contained in the ideal generated by a given set of polynomials. We define and construct such subideal border bases, provide som
Here we study the problem of generalizing one of the main tools of Groebner basis theory, namely the flat deformation to the leading term ideal, to the border basis setting. After showing that the straightforward approach based on the deformation to
The main topic of the paper is the construction of various explicit flat families of border bases. To begin with, we cover the punctual Hilbert scheme Hilb^mu(A^n) by border basis schemes and work out the base changes. This enables us to control flat