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In this paper we study the bivariate truncated moment problem (TMP) on curvesof the form $y=q(x)$, $q(x)\in \mathbb{R}[x]$, $\text{deg } q\geq 3$, and$yx^\ell=1$, $\ell\in \mathbb{N}\setminus\{1\}$. For even degree sequences thesolution based on the number of moment matrix extensions was first given byFialkow using the truncated Riesz-Haviland theorem and a sum-of-squaresrepresentations for polynomials, strictly positive on such curves. Namely, theupper bound on this number is quadratic in the degrees of the sequence and thepolynomial determining a curve. We use a reduction to the univariate settingtechnique and improve Fialkow's bound to $\text{deg }q-1$ (resp. $\ell+1$) forcurves $y=q(x)$ (resp. $yx^\ell=1$). This in turn gives analogous improvementsof the degrees in the sum-of-squares representations referred to above.Moreover, we get the upper bounds on the number of atoms in the minimalrepresenting measure, which are $k\;\text{deg }q$ (resp. $k(\ell+1)$) forcurves $y=q(x)$ (resp. $yx^\ell=1$) for even degree sequences, while for oddones they are $k\;\text{deg }q-\big\lceil\frac{\text{deg }q}{2} \big\rceil$(resp. $k(\ell+1)-\big\lfloor\frac{\ell}{2} \big\rfloor+1$) for curves $y=q(x)$(resp. $yx^\ell=1$). In the even case these are counterparts to the result byRiener and Schweighofer, which gives the same bound for odd degree sequences onall plane curves, while in the odd case it is a slight improvement of theirbound in these special cases. Further on, we give another solution to the TMPon the curves studied based on the feasibility of a linear matrix inequality,corresponding to the univariate sequence obtained, and finally we solveconcretely odd degree cases of the TMP on curves $y=x^\ell$, $\ell=2,3$, andadd a new solvability condition to the even degree case on the curve $y=x^2$.

The truncated moment problem on curves $y=q(x)$ and $yx^\ell=1$