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Concrete Solution to the Nonsingular Quartic Binary Moment Problem

Given real numbers $β\equiv β^{\left( 4\right) }\colon β_{00}$, $β_{10}$, $β_{01}$, $β_{20}$, $β_{11}$, $ β_{02}$, $β_{30}$, $β_{21}$, $β_{12}$, $β_{03}$, $β_{40}$, $β_{31}$, $β_{22}$, $β_{13}$, $β_{04}$, with $β_{00} >0$, the quartic real moment problem for $β$ entails finding conditions for the existence of a positive Borel measure $μ$, supported in $\mathbb{R}^2$, such that $β_{ij}=\int s^{i}t^{j}\,dμ\;\;(0\leq i+j\leq 4) $. Let $\mathcal{M}(2)$ be the 6 x 6 moment matrix for $β^{(4)}$, given by $\mathcal{M}(2)_{\mathbf{i},\mathbf{j}}:=β_{\mathbf{i}+\mathbf{j}}$, 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 $β^{(4)}$ when $\mathcal{M}(2)$ is nonsingular; moreover, we prove that it is possible to ensure that one such representing measure is 6-atomic.