Degree k Column Relations for Moment Matrices

Marc Moore

Let M(k) be the moment matrix associated with a complex moment sequence γ^(2k). Let V ≡ V (M(k)) be the algebraic variety associated with M(k). That is, if p(z, z) is a polynomial which corresponds to a column relation of M(k), then V (M(k)) is the intersection of the zero locus of p(z, z) and p(z, z). If the rank of M(k) is equal to the cardinality of V , we say that the problem is extremal. In this talk, we show that if G is the Groebner basis for the ideal I(V ) associated with V by the Nullstellensatz, then G contains exactly the polynomials which correspond to the rest of the column relations of M(k). Moreover, we can find a numerical condition on the level of the moments which is equivalent to the existence of a representing measure.

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