The moment problem on curves with bumps

TL;DR

This paper explores the power moment problem on certain curves with bumps, aiming to extend non-negativity criteria to more complex geometric shapes in higher dimensions.

Contribution

It combines known cases of curves and semi-algebraic sets to broaden the class of shapes where the moment problem can be solved via non-negativity certificates.

Findings

Extended moment problem solutions to new classes of curves with bumps.

Identified conditions under which non-negativity certificates apply.

Enhanced understanding of geometric configurations supporting moment measures.

Abstract

The power moments of a positive measure on the real line or the circle are characterized by the non-negativity of an infinite matrix, Hankel, respectively Toeplitz, attached to the data. Except some fortunate configurations, in higher dimensions there are no non-negativity criteria for the power moments of a measure to be supported by a prescribed closed set. We combine two well studied fortunate situations, specifically a class of curves in two dimensions classified by Scheiderer and Plaumann, and compact, basic semi-algebraic sets, with the aim at enlarging the realm of geometric shapes on which the power moment problem is accessible and solvable by non-negativity certificates.