Cubic column relations in truncated moment problems

For the truncated moment problem associated to a complex sequence γ(2n)={γij}i,j∈Z+,i+j⩽2n to have a representing measure μ  , it is necessary for the moment matrix M(n)M(n) to be positive semidefinite, and for the algebraic variety VγVγ to satisfy rankM(n)⩽cardVγ as well as a consistency condition: the Riesz functional vanishes on every polynomial of degree at most 2n   that vanishes on VγVγ. In previous work with L. Fialkow and H.M. Möller, the first named author proved that for the extremal case (rankM(n)=cardVγ), positivity and consistency are sufficient for the existence of a representing measure. In this paper we solve the truncated moment problem for cubic   column relations in M(3)M(3) of the form Z3=itZ+uZ¯ (u,t∈Ru,t∈R); we do this by checking consistency. For (u,t)(u,t) in the open cone determined by 0<|u|