Semidefinite Characterization and Computation of Real Radical Ideals

TL;DR

This paper introduces a semidefinite approach to characterize and compute the real radical of zero-dimensional ideals in real polynomial rings, using moment relaxations and numerical methods.

Contribution

It presents a novel semidefinite characterization of the real radical ideal and an algorithm that efficiently computes real solutions and generators without complex component computation.

Findings

Successfully characterizes real radical ideals via semidefinite programming.

Provides an algorithm to compute all real solutions of zero-dimensional ideals.

Generates generators of the real radical as border or Gröbner bases.

Abstract

For an ideal $I\subseteq\mathbb{R}[x]$ given by a set of generators, a new semidefinite characterization of its real radical $I(V_\mathbb{R}(I))$ is presented, provided it is zero-dimensional (even if $I$ is not). Moreover we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute all (finitely many) points of the real variety $V_\mathbb{R}(I)$ as well as a set of generators of the real radical ideal. The latter is obtained in the form of a border or Gr\"obner basis. The algorithm is based on moment relaxations and, in contrast to other existing methods, it exploits the real algebraic nature of the problem right from the beginning and avoids the computation of complex components.