The role of Gröbner 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 ∈ Z + ), with column relations given by general harmonic polynomials. We accomplish this by proving that the Gröbner 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.