Almost Vanishing Polynomials for Sets of Limited Precision Points
For researchers in numerical algebraic geometry and data analysis, this work provides a method to handle uncertainty in point coordinates when computing vanishing polynomials, which is an incremental improvement over existing approaches.
The paper addresses the problem of computing polynomials that almost vanish on a set of imprecise points and all its equivalent sets within a given tolerance. It presents an algorithm that produces a set of such polynomials, which can characterize the geometric configuration of the points despite data uncertainty.
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*.