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

TL;DR

This paper advances the understanding of the truncated moment problem on specific algebraic curves by providing improved bounds on solutions, minimal measures, and new solvability conditions, using reduction techniques and sum-of-squares representations.

Contribution

It introduces improved bounds for the TMP on curves y=q(x) and yx^l=1, and offers new solutions based on linear matrix inequalities, refining previous results.

Findings

Improved bounds on the number of moment matrix extensions.

Tighter bounds on the minimal number of atoms in representing measures.

New solvability conditions for specific cases of the TMP.

Abstract

In this paper we study the bivariate truncated moment problem (TMP) on curves of 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 the solution based on the number of moment matrix extensions was first given by Fialkow using the truncated Riesz-Haviland theorem and a sum-of-squares representations for polynomials, strictly positive on such curves. Namely, the upper bound on this number is quadratic in the degrees of the sequence and the polynomial determining a curve. We use a reduction to the univariate setting technique and improve Fialkow's bound to $\text{deg }q-1$ (resp. $\ell+1$) for curves $y=q(x)$ (resp. $yx^\ell=1$). This in turn gives analogous improvements of the degrees in the sum-of-squares representations referred to above. Moreover, we get the upper bounds on the number of atoms in the minimal representing measure, which are $k\;\text{deg }q$ (resp. $k(\ell+1)$) for curves $y=q(x)$ (resp. $yx^\ell=1$) for even degree sequences, while for odd ones 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 by Riener and Schweighofer, which gives the same bound for odd degree sequences on all plane curves, while in the odd case it is a slight improvement of their bound in these special cases. Further on, we give another solution to the TMP on the curves studied based on the feasibility of a linear matrix inequality, corresponding to the univariate sequence obtained, and finally we solve concretely odd degree cases of the TMP on curves $y=x^\ell$, $\ell=2,3$, and add a new solvability condition to the even degree case on the curve $y=x^2$.