Concrete solution to the nonsingular quartic binary moment problem

Abstract:

Given real numbers $\beta \equiv \beta ^{\left ( 4\right ) }\colon \beta _{00}$, $\beta _{10}$, $\beta _{01}$, $\beta _{20}$, $\beta _{11}$, $\beta _{02}$, $\beta _{30}$, $\beta _{21}$, $\beta _{12}$, $\beta _{03}$, $\beta _{40}$, $\beta _{31}$, $\beta _{22}$, $\beta _{13}$, $\beta _{04}$, with $\beta _{00} >0$, the quartic real moment problem for $\beta$ entails finding conditions for the existence of a positive Borel measure $\mu$, supported in $\mathbb {R}^2$, such that $\beta _{ij}=\int s^{i}t^{j} d\mu \;\;(0\leq i+j\leq 4)$. Let $\mathcal {M}(2)$ be the $6 \times 6$ moment matrix for $\beta ^{(4)}$, given by $\mathcal {M}(2)_{\mathbf {i}, \mathbf {j}}:=\beta _{\mathbf {i}+\mathbf {j}}$, where $\mathbf {i},\mathbf {j} \in \mathbb {Z}^2_+$ and $\left |\mathbf {i}\right |,\left |\mathbf {j}\right |\le 2$. 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.