An Approximation to the Gröbner Basis of Ideals of Perturbed Points: Part I

We develop a method for approximating the Gröbner basis of the ideal of polynomials which vanish at a finite set of points, when the coordinates of the points are known with only limited precision. The method consists of a preprocessing phase of the input points to mitigate the effects of the input data uncertainty, and of a new “numerical” version of the Buchberger-Möller algorithm to compute an approximation GB to the exact Gröbner basis. This second part is based on a threshold-dependent procedure for analyzing from a numerical point of view the membership of a perturbed vector to a perturbed subspace. With a suitable choice of the threshold, the set GB turns out to be a good approximation to a “possible” exact Gröbner basis or to a basis which is an “attractor” of the exact one. In addition, the polynomials of GB are “sufficiently near” to the polynomials of the extended basis, introduced by Stetter, but they present the advantage that LT (GB) coincides with the leading terms of a “possible” exact case. The set of the preprocessed points, approximation to the unknown exact points, is a pseudozero set for the polynomials of GB.