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

On a minimal solution for the indefinite truncated multidimensional moment problem