Almost Vanishing Polynomials for Sets of Limited Precision Points

TL;DR

This paper introduces an algorithm to compute polynomials that approximately vanish on a set of points with limited precision, capturing the underlying geometric configuration despite data uncertainties.

Contribution

The paper presents a novel algorithm for finding almost vanishing polynomials that characterize a point set and its equivalents, robust to data errors.

Findings

Algorithm effectively identifies polynomials that nearly vanish on uncertain data.

The method captures the geometric configuration of points despite limited precision.

It provides a stable alternative to traditional vanishing ideal bases affected by data uncertainty.

Abstract

Let X be a set of s points whose coordinates are known with only limited From the numerical point of view, given a set X of s real points whose coordinates are known with only limited precision, each set X* of real 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 Tol on the data error, computes a set G of polynomials such that each element of G "almost vanishing" at X and at all its equivalent sets X*. Even if G is not, in the general case, a basis of the vanishing ideal I(X), we show that, differently from the basis of I(X) that can be greatly influenced by the data uncertainty, G can determine a geometrical configuration simultaneously characterizing the set X and all its equivalent sets X*.