Non-extremal sextic moment problems

TL;DR

This paper addresses non-extremal sextic moment problems, providing solutions for specific rank and variety cases by extending rank-reduction techniques to construct representing measures.

Contribution

It settles three key instances of non-extremal sextic moment problems, introducing algorithms and generalizing previous rank-reduction methods.

Findings

Guarantees 7-atomic measure when r=7, v=9, positive semidefiniteness, consistency, and variety condition hold.

Constructs two algorithms for r=8, v=9 and v=∞ cases.

Extends rank-reduction techniques to higher-order moment problems.

Abstract

Positive semidefiniteness, recursiveness, and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic and quartic moment problems. Also, positive semidefiniteness, combined with another necessary condition, consistency, is a sufficient condition in the case of extremal moment problems, i.e., when the rank of the moment matrix (denoted by r) and the cardinality of the associated algebraic variety (denoted by v) are equal. However, these conditions are not sufficient for non-extremal sextic or higher-order truncated moment problems. In this paper we settle three key instances of the non-extremal (i.e., r<v) sextic moment problem, as follows: when r=7, positive semidefiniteness, consistency and the variety condition guarantee the existence of a 7-atomic representing measure; when r=8 we construct two determining algorithms, corresponding to the cases v=9 and $v=+ \infty$. \ To accomplish this, we generalize the rank-reduction technique developed in previous work, where we solved the nonsingular quartic moment problem and found an explicit way to build a representing measure.