Stable Border Bases for Ideals of Points

TL;DR

This paper introduces a method to compute stable polynomial bases for the vanishing ideal of a set of points, ensuring robustness against small data perturbations in the points' coordinates.

Contribution

It presents a novel approach to obtain structurally stable border bases for ideals of points, independent of data uncertainty.

Findings

The method produces bases resilient to small data perturbations.

The approach maintains structural similarity of bases under data noise.

It enhances the robustness of polynomial ideal computations.

Abstract

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)$.