On a minimal solution for the indefinite truncated multidimensional moment problem

TL;DR

This paper characterizes the minimal support size for signed measures representing indefinite truncated multidimensional moments, using rank-preserving extensions of Hankel matrices and real radical polynomial ideals.

Contribution

It provides necessary and sufficient conditions for minimal signed representing measures in the indefinite multidimensional moment problem, extending previous results to a broader class.

Findings

Conditions for minimal support signed measures are established.

Existence of rank-preserving Hankel matrix extensions is crucial.

Illustrations on concrete examples demonstrate the theory.

Abstract

We will consider the indefinite truncated multidimensional moment problem. Necessary and sufficient conditions for a given truncated multisequence to have a signed representing measure $\mu$ with ${\rm card}\,{\rm supp}\, \mu$ as small as possible are given by the existence of a rank preserving extension of a multivariate Hankel matrix (built from the given truncated multisequence) such that the corresponding associated polynomial ideal is real radical. This result is a special case of a more general characterisation of truncated multisequences with a minimal complex representing measure whose support is symmetric with respect to complex conjugation (which we will call {\it quasi-complex}). One motivation for our results is the fact that positive semidefinite truncated multisequence need not have a positive representing measure. Thus, our main result gives the potential for computing a signed representing measure $\mu = \mu_+ - \mu_-$, where ${\rm card} \,\mu_-$ is small. We illustrate this point on concrete examples.