The Role of Gr\"obner Bases in the Study of Extremal Truncated Moment Problems

TL;DR

This paper generalizes a 2014 result on the existence of representing measures for moment matrices with harmonic polynomial relations, using Gr"obner bases to characterize necessary and sufficient conditions.

Contribution

It extends the 2014 findings to arbitrary moment matrices with harmonic polynomial relations, employing Gr"obner bases for a comprehensive characterization.

Findings

Gr"obner basis provides all necessary column relations

Characterization of representing measure conditions

Generalization to arbitrary moment matrices

Abstract

In a 2014 paper, R.E. Curto and S. Yoo proved that a moment matrix $M(3)$ with specific harmonic polynomials as column relations admits a representing measure if and only if a condition at the level of moments holds. \ In this paper, we generalize the 2014 result to arbitrary moment matrices $M(k)$ ($k \in \mathbb{Z}_{+}$), with column relations given by general harmonic polynomials. \ We accomplish this by proving that the Gr\"obner basis for the ideal generated by a finite variety associated with the moment matrix provides all the necessary column relations for the matrix as well as a suitable condition on the moments, which is equivalent to the existence of a representing measure.