On a minimal solution for the indefinite truncated multidimensional moment problem

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 $\mathrm{card}\mathrm{supp}\mu$ 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 $\mathrm{card}{\mu }_{-}$ is small. We illustrate this point on concrete examples.