On the Minimal Number of Moments Required to Recover the Sublevel Set of a Homogeneous Polynomial

In this paper we study a problem posed by Jean B. Lasserre in \cite{La}. Namely, if $G =\{x\in \zR^n : g(x)\le 1\}$ is the compact sublevel set of a non-negative $k$-homogeneous polynomial $g$ of $n$ variables, we seek to find the minimum number of moments required to determine or to recover $g.$ For polynomials of degree two, we can recover $g$ computing eigenvalues and eigenvectors associated to a matrix $M \in \mathbb R^{n \times n}$.For polynomials of degree $k$, we only prove that $g$ is determined by all its $\binom{n+k-1}{k}$ moments of degree $k$