SOLUTION OF THE SINGULAR QUARTIC MOMENT PROBLEM

In this note we obtain a complete solution to the quartic problem in the case when the associated moment matrix M(2)(γ) is singular. Each representing measure μ satisfies card supp μ ≥ rank M(2), and we develop concrete necessary and sufficient conditions for the existence and uniqueness of representing measures, particularly minimal ones. We show that rank M(2)-atomic minimal representing measures exist in case the moment problem is subordinate to an ellipse or non-degenerate hyperbola. If the quartic moment problem is subordinate to a pair of intersecting lines, those problems subordinate to a general intersection of two conics may not have any representing measure at all. As an application, we describe the minimal quadrature rules of degree 4 for arclength measure on a parabolic arc.