Abstract: | Let denote a sequence of complex numbers ( ), and let denote a closed subset of the complex plane . The Truncated Complex -Moment Problem for entails determining whether there exists a positive Borel measure on such that ( ) and . For a semi-algebraic set determined by a collection of complex polynomials , 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 and the localizing matrices . We prove that there exists a -atomic representing measure for supported in if and only if and there is some rank-preserving extension for which , where or . |