An approximation of the Gr\"obner basis of ideals of perturbed points, part I

TL;DR

This paper introduces a method to approximate the Gr"obner basis of polynomial ideals for perturbed points with limited precision, combining preprocessing and a numerical Buchberger-M"oller algorithm to handle data uncertainty.

Contribution

It presents a novel numerical approach for approximating Gr"obner bases of ideals of perturbed points, incorporating data preprocessing and threshold-based analysis.

Findings

The method effectively approximates the Gr"obner basis under data perturbations.

The approach maintains the leading terms consistent with an exact basis.

Preprocessed points form a pseudozero set for the approximated basis.

Abstract

We develop a method for approximating the Gr\"obner 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-M\"oller algorithm to compute an approximation $\bar{GB}$ to the exact Gr\"obner 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 $\bar{GB}$ turns out to be a good approximation to a "possible" exact Gr\"obner basis or to a basis which is an "attractor" of the exact one. In addition, the polynomials of $\bar{GB}$ are "sufficiently near" to the polynomials of the extended basis, introduced by Stetter, but they present the advantage that $LT(\bar{GB})$ 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 $\bar{GB}$.