The Division Algorithm in Sextic Truncated Moment Problems

Abstract

For a degree 2n finite sequence of real numbers $\beta \equiv \beta^{(2n)}= \{ \beta_{00},\beta_{10}, \beta_{01},\cdots, \beta_{2n,0}, \beta_{2n-1,1},\cdots, \beta_{1,2n-1},\beta_{0,2n} \}$ to have a representing measure $\mu $, it is necessary for the associated moment matrix $\mathcal{M}(n)$ to be positive semidefinite, and for the algebraic variety associated to $\beta $, $\mathcal{V}_{\beta} \equiv \mathcal{V}(\mathcal{M}(n))$, to satisfy $\operatorname{rank} \mathcal{M}(n)\leq \operatorname{card} \mathcal{V}_{\beta}$ as well as the following consistency} condition: if a polynomial $p(x,y)\equiv \sum_{ij}a_{ij}x^{i}y^j$ of degree at most 2n vanishes on $\mathcal{V}_{\beta}$, then the Riesz functional $\Lambda (p) \equiv p(\beta ):=\sum_{ij}a_{ij}\beta _{ij}=0$. 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 consistency, is a sufficient condition in the case of 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. For extremal sextic moment problems, verifying consistency amounts to having good representation theorems for sextic polynomials in two variables vanishing on the algebraic variety of the moment sequence. We obtain such representation theorems using the Division Algorithm from algebraic geometry. As a consequence, we are able to complete the analysis of extremal sextic moment problems.