Concrete Solution to the Nonsingular Quartic Binary Moment Problem

TL;DR

This paper provides explicit solutions for the nonsingular quartic binary moment problem, demonstrating the existence of a 6-atomic representing measure under certain conditions.

Contribution

It offers a concrete method to find representing measures for nonsingular moment matrices in the quartic binary case, including the construction of a 6-atomic measure.

Findings

Explicit representing measures are constructed for nonsingular moment matrices.

A 6-atomic measure can always be ensured as a solution.

The approach applies to the quartic binary moment problem with positive definite moment matrices.

Abstract

Given real numbers $\beta \equiv \beta ^{\left( 4\right) }\colon \beta_{00}$, $\beta _{10}$, $\beta _{01}$, $\beta _{20}$, $\beta _{11}$, $ \beta _{02}$, $\beta _{30}$, $\beta _{21}$, $\beta _{12}$, $\beta _{03}$, $\beta _{40}$, $\beta _{31}$, $\beta _{22}$, $\beta _{13}$, $\beta _{04}$, with $\beta _{00} >0$, the quartic real moment problem for $\beta $ entails finding conditions for the existence of a positive Borel measure $\mu $, supported in $\mathbb{R}^2$, such that $\beta _{ij}=\int s^{i}t^{j}\,d\mu \;\;(0\leq i+j\leq 4) $. Let $\mathcal{M}(2)$ be the 6 x 6 moment matrix for $\beta^{(4)}$, given by $\mathcal{M}(2)_{\mathbf{i},\mathbf{j}}:=\beta_{\mathbf{i}+\mathbf{j}}$, where $\mathbf{i},\mathbf{j} \in \mathbb{Z}^2_+$ and $\left|\mathbf{i}\right|,\left|\mathbf{j}\right|\le 2$. In this note we find concrete representing measures for $\beta^{(4)}$ when $\mathcal{M}(2)$ is nonsingular; moreover, we prove that it is possible to ensure that one such representing measure is 6-atomic.