A constructive approach to the truncated moment problem on cubic curves in Weierstrass form

TL;DR

This paper presents a constructive solution to the truncated moment problem on cubic curves in Weierstrass form, establishing conditions for the existence of atomic measures with minimal support.

Contribution

It extends the TMP solution to a class of cubic curves, providing constructive methods and numerical examples, including symmetric cases with vanishing odd moments.

Findings

Existence of rank-attaining atomic measures on certain cubic curves.

Constructive solution for symmetric cases with vanishing odd moments.

Numerical example requiring one more atom than the rank of the moment matrix.

Abstract

In this paper, we develop a constructive solution for the pure truncated moment problem on cubic curves in Weierstrass form, establishing the existence of a representing measure whose number of atoms equals the rank of the associated moment matrix. By a recent result of Baldi, Blekherman, and Sinn, for projectively smooth curves whose projective closure has exactly one real point at infinity, the existence of such a rank-attaining atomic measure is equivalent to the existence of a representing measure; consequently, the TMP is constructively solved for this class of curves. We also present a numerical degree--$6$ example in which every minimal representing measure supported on the cubic curve requires $\operatorname{rank} M(3)+1$ atoms, where $M(3)$ denotes the moment matrix. Finally, we provide a constructive solution for the symmetric case, i.e., when all moments of odd degree in $y$ vanish.