On the Closure of Positive Flat Moment Matrices

Let y ≡ y(2d) = {yi}i∈Zn+ ,|i|62d denote a real n-dimensional multisequence of degree 2d, y0 > 0. Let Ly : R2d[x1, . . . , xn] 7→ R denote the Riesz functional, defined by Ly(∑|i|62d aix i) = ∑ aiyi, and let Md(y) denote the corresponding moment matrix. Positivity of Ly plays a significant role in the Truncated Moment Problem and in the Polynomial Optimization Problem, but concrete conditions for positivity are unknown in general. Md(y) is flat if rankMd(y) = rankMd−1(y); it is known that if Md(y) is positive semidefinite and flat, then y has a representing measure (and Ly is positive). Let Fd := {y ≡ y(2d) : Md(y) 0 is flat}. If y ∈ F d (the closure), then y does not necessarily have a representing measure, but Ly is positive, so Md(y) 0 and, moreover, rankMd(y) 6 dimRd−1[x1, . . . , xn]. We prove, conversely, that these positivity and rank conditions for Md(y) are sufficient for membership in F d in two basic cases: when n = 1, d > 1, and when n = d = 2.