An SDP-based Method for the Real Radical Ideal Membership Test

Let V be the set of real solutions of a system of multivariate polynomial equations with real coefficients. The real radical ideal (RRI) of V is the infinite set of multivariate polynomials that vanish on V . We give theoretical results that yield a finite step numerical algorithm for testing if a given polynomial is a member of this RRI. The paper exploits recent work that connects solution sets of such real polynomial systems with solution sets of semidefinite programming, SDP, problems involving moment matrices. We take advantage of an SDP technique called facial reduction. This technique regularizes our problem by projecting the feasible set onto the so-called minimal face. In addition, we use the Douglas-Rachford iterative approach which has advantages over traditional interior point methods for our application. If V has finitely many real solutions, then our method yields a finite set of polynomials in the form of a geometric involutive basis that are generators of the RRI and form an RRI membership test. In the case where the set V has real solution components of positive dimension, and given an input polynomial of degree d, our method can also decide RRI membership via a truncated geometric involutive basis of degree d. Examples are given to illustrate our approach and its advantages that remove multiplicities and sums of squares that cause illconditioning for real solutions of polynomial systems.