Revisiting two theorems of Curto and Fialkow on moment matrices

We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence $y\in \mathbb{R}^{\mathbb{Z}^n_+}$whose moment matrix $M(y)$ is positive semidefinite and has finite rank $r$ is the sequence of moments of an $r$-atomic nonnegative measure $\mu$on $\mathbb{R}^n$. We give an alternative proof for this result, using algebraic tools (the Nullstellensatz) in place of the functional analytic tools used in the original proof of Curto and Fialkow. An easy observation is the existence of interpolation polynomials at the atoms of the measure $\mu$having degree at most $t$ if the principal submatrix $M_t(y)$of $M(y)$ (indexed by all monomials of degree $\le t$) has full rank $r$. This observation enables us to shortcut the proof of the following result. Consider a basic closed semialgebraic set $F=\{x\in \mathbb{R}^n\mid h_1(x)\ge 0, \ldots,h_m(x)\ge 0\}$, where $h_j\in \mathbb{R}[x_1,\ldots,x_n]$ and $d:=\operatorname{max}_{j=1}^m \lceil \operatorname{deg}(h_j)/2\rceil$. If $M_t(y)$ is positive semidefinite and has a flat extension $M_{t+d}(y)$ such that all localizing matrices $M_{t}(h_j\ast y)$ are positive semidefinite, then $y$ has an atomic representing measure supported by $F$. We also review an application of this result to the problem of minimizing a polynomial over the set $F$.