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

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.