A Certificate for Semidefinite Relaxations in Computing Positive Dimensional Real Varieties

For an ideal I with a positive dimensional real variety VR(I), based on moment relaxations, we study how to compute a Pommaret basis which is simultaneously a Gröbner basis of an ideal J generated by the kernel of a truncated moment matrix and satisfying I ⊆ J ⊆ I(VR(I)), VR(I) = VC(J) ∩ R. We provide a certificate consisting of a condition on coranks of moment matrices for terminating the algorithm. For a generic δ-regular coordinate system, we prove that the condition is satisfiable in a large enough order of moment relaxations.