The truncated complex K-moment problem

Let γ ≡ γ ( 2n ) denote a sequence of complex numbers γ ₀₀, γ ₀₁, γ ₁₀, , γ 0,2n, , γ 2n,0 (γ ₀₀>0, γ ᵢⱼ=γ ̄ⱼᵢ), and let K denote a closed subset of the complex plane C. The Truncated Complex K-Moment Problem for γ entails determining whether there exists a positive Borel measure μ on C such that γ ᵢⱼ=∫ z̄ⁱzʲ dμ (0≤ i+j≤ 2n) and suppμ ⊆ K. For K≡ KP a semi-algebraic set determined by a collection of complex polynomials P = \ pᵢ ( z,z̄ ) \ ᵢ₌₁ᵐ, 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 ) ( γ ) and the localizing matrices M_pᵢ. We prove that there exists a rankM ( n )-atomic representing measure for γ ( 2n ) supported in KP if and only if M ( n ) ≥ 0 and there is some rank-preserving extension M ( n+1 ) for which M_pᵢ ( n+kᵢ ) ≥ 0, where pᵢ=2kᵢ or 2kᵢ-1 (1≤ i≤ m).