A certificate for semidefinite relaxations in computing positive-dimensional real radical ideals

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)∩Rn. 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.