The truncated complex K-moment problem

Let ^ (2n) denote a sequence of complex numbers ₀₀, ₀₁, ₁₀, , ₀, ₂₍, , ₂₍, ₀ (₀₀>0, ₈₉= ₉₈), and let K denote a closed subset of the complex plane C. The Truncated Complex K-Moment Problem for entails determining whether there exists a positive Borel measure on C such that ₈₉= z^iz^j d (0 i+j 2n) and supp K. For K K ₏ a semi-algebraic set determined by a collection of complex polynomials P = \ p₈ (z, {z) \} ₈=₁^m, we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix M (n) () and the localizing matrices M₏_₈. We prove that there exists a rankM (n) -atomic representing measure for ^ (2n) supported in K ₏ if and only if M (n) 0 and there is some rank-preserving extension M (n+1) for which M₏_₈ (n+k₈) 0, where p₈=2k₈ or 2k₈-1 (1 i m).