Simple Varieties for Limited Precision Points

Given a finite set X of points and a tolerance epsilon representing the maximum error on the coordinates of each point, we address the problem of computing a simple polynomial f whose zero-locus Z(f) ``almost'' contains the points of X. We propose a symbolic-numerical method that, starting from the knowledge of X and epsilon, determines a polynomial f whose degree is strictly bounded by the minimal degree of the lements of the vanishing ideal of X. Then we state the sufficient conditions for proving that Z(f) lies close to each point of X by less than epsilon. The validity of the proposed method relies on a combination of classical results of Computer Algebra and Numerical Analysis; its effectiveness is illustrated with a number of examples.