Solution of the truncated hyperbolic moment problem

Let Q(x,y)=0 be an hyperbola in the plane. Given real numbers $\beta \equiv\beta^{2n)}={\beta_{ij}}{i,j\geq0,i+j\leq2n}$,with$\beta{00}>0$, the truncated Q-hyperbolic moment problem for \beta entails finding necessary and sufficient conditions for the existence of a positive Borel measure \mu, supported in Q(x,y)=0, such that $\beta_{ij}=\int y^{i}x^{j} d\mu (0\leq i+j\leq2n)$. We prove that \beta admits a Q-representing measure \mu (as above) if and only if the associated moment matrix $\mathcal{M}(n)(\beta )$ is positive semidefinite, recursively generated, has a column relation Q(X,Y)=0, and the algebraic variety $\mathcal{V}(\beta )$ associated to \beta satisfies $card\mathcal{V}(\beta )\ge \text{rank}\mathcal{M}(n)(\beta )$. In this case, $rank\mathcal{M}(n)\le 2n+1$; if $rank\mathcal{M}(n)\le 2n$, then \beta admits a $rank\mathcal{M}(n)$-atomic (minimal) Q-representing measure; if $rank\mathcal{M}(n)=2n+1$, then \beta admits a Q-representing measure \mu satisfying $2n+1\text{leqcard}supp\mu \le 2n+2$.