Verified error bounds for real solutions of positive-dimensional polynomial systems

In this paper, we propose two algorithms for verifying the existence of real solutions of positive-dimensional polynomial systems. The first one is based on the critical point method and the homotopy continuation method. It targets for verifying the existence of real roots on each connected component of an algebraic variety V ∩ Rn defined by polynomial equations. The second one is based on the low-rank moment matrix completion method and aims for verifying the existence of at least one real roots on V ∩ Rn. Combined both algorithms with the verification algorithms for zero-dimensional polynomial systems, we are able to find verified real solutions of positive-dimensional polynomial systems very efficiently for a large set of examples.