Extension of the Nonsingular Quartic Moment Problem

In this paper,Consider the quartic moment problem suggested by Curto and Fialkow,Given complex numbersγ≡γ~((4)):γ_(00),γ_(01),γ_(10),γ_(02),γ_(11),γ_(20),γ_(03),γ_(12),γ_(21),γ_(30),γ_(04),γ_(13),γ_(22),γ_(31),γ_(40),withγ_(00)0 andγ_(ij)=γ_(ji),Find a positive Borel measureμsuch thatγ_(ij)=∫Z~iZ~jdμ,(0≤i + j≤4),Examine the existence of flat extensions for nonsingular positive quartic moment matrices M(2),Give partial solutions for the nonsingular quartic moment problem and examples are provided.