Concrete Solution to the Nonsingular Quartic Binary Moment Problem

Given real numbers $\beta \equiv {\beta }^{\left(4\right)}\text{\hspace{0.17em}⁣}:{\beta }_{00}$, ${\beta }_{10}$, ${\beta }_{01}$, ${\beta }_{20}$, ${\beta }_{11}$, ${\beta }_{02}$, ${\beta }_{30}$, ${\beta }_{21}$, ${\beta }_{12}$, ${\beta }_{03}$, ${\beta }_{40}$, ${\beta }_{31}$, ${\beta }_{22}$, ${\beta }_{13}$, ${\beta }_{04}$, with $\beta _{00}

0$,thequarticrealmomentproblemfor$\beta $ entails finding conditions for the existence of a positive Borel measure $\mu$, supported in ${\mathbb{R}}^2$, such that $\beta _{ij}=\int s^{i}t^{j},d\mu ;;(0\leq i+j\leq

4. $\mathrm{.}Let$$\mathcal{M}$(2)$bethe6x6momentmatrixfor$\beta^{(4)}$, given by $\mathcal{M}(2{)}_{\mathbf{i},\mathbf{j}}:={\beta }_{\mathbf{i}+\mathbf{j}}$, where $\mathbf{i},\mathbf{j}\in {\mathbb{Z}}_{+}^2$ and $\mid \mathbf{i}\mid ,\mid \mathbf{j}\mid \le 2$. In this note we find concrete representing measures for ${\beta }^{(4)}$ when $\mathcal{M}(2)$ is nonsingular; moreover, we prove that it is possible to ensure that one such representing measure is 6-atomic.