The truncated complex K-moment problem

Let gamma = gamma((2n)) denote a sequence of complex numbers gamma(00), gamma(01), gamma(10), ..., gamma(0,2n), ..., gamma(2n,0) (gamma(00) > 0, gamma(ij) = <(gamma)over bar>(ji)), and let K denote a closed subset of the complex plane C. The Truncated Complex K-Moment Problem for gamma entails determining whether there exists a positive Borel measure mu on C such that gamma(ij) = integral (z) over bar(i)z(j) d mu (0 less than or equal to i + j less than or equal to 2n) and supp mu subset of or equal to K. For K = K-P a semi-algebraic set determined by a collection of complex polynomials P = {p(i) (z, (z) over bar)}(i=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) (gamma) and the localizing matrices M-pi. We prove that there exists a rank M (n)-atomic representing measure for gamma((2n)) supported in K-P if and only if M (n) greater than or equal to 0 and there is some rank-preserving extension M (n + 1) for which M-pi (n + k(i)) greater than or equal to 0, where deg p(i) = 2k(i) or 2k(i) - 1 (1 less than or equal to i less than or equal to m).