An algorithmic approach to the sextic moment problem

We describe an approach to solving the sextic moment problem based on a thorough analysis of the extremal case, and algorithmic solutions for the non-extremal cases. \medskip For a degree $2n$ complex sequence $\gamma \equiv \gamma ^{(2n)}=\{\gamma _{ij}\}_{i,j\in Z_{+},i+j \leq 2n}$ to have a representing measure $\mu $, it is necessary for the associated moment matrix $M(n)$ to be positive semidefinite, and for the algebraic variety associated to $\gamma $, $\mathcal{V}_{\gamma} \equiv \mathcal{V}(M(n))$, to satisfy rank $M(n)\leq \;$ card$\;\mathcal{V}_{\gamma}$ as well as the following consistency condition: if a polynomial $p(z,\bar{z})\equiv \sum_{ij}a_{ij}\bar{z}^{i}z^j$ of degree at most $2n$ vanishes on $\mathcal{V}_{\gamma}$, then the \textit{Riesz functional} $\Lambda (p) \equiv p(\gamma ):=\sum_{ij}a_{ij}\gamma _{ij}=0$. \medskip Positive semidefiniteness, recursiveness, and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic ($n=1$) and quartic ($n=2$) moment problems. \ Also, positive semidefiniteness, combined with the above mentioned consistency condition, is a sufficient condition in the case of \textit{extremal} moment problems, i.e., when the rank of the moment matrix (denoted by $r$) and the cardinality of the associated algebraic variety (denoted by $v$) are equal. \ However, these conditions are not sufficient for \textit{non}-extremal (i.e., $