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We investigate and discuss when the inverse of a multivariate truncatedmoment matrix of a measure $\mu$ has zeros in some prescribed entries. Wedescribe precisely which pattern of these zeroes corresponds to independence,namely, the measure having a product structure. A more refined finding is thatthe key factor forcing a zero entry in this inverse matrix is a certainconditional triangularity property of the orthogonal polynomials associatedwith $\mu$.

Measures with zeros in the inverse of their moment matrix