Reconstruction of algebraic-exponential data from moments

Let $G$ be a bounded open subset of Euclidean space with real algebraic boundary $Γ$. Under the assumption that the degree $d$ of $Γ$ is given, and the power moments of the Lebesgue measure on $G$ are known up to order $3d$, we describe an algorithmic procedure for obtaining a polynomial vanishing on $Γ$. The particular case of semi-algebraic sets defined by a single polynomial inequality raises an intriguing question related to the finite determinateness of the full moment sequence. The more general case of a measure with density equal to the exponential of a polynomial is treated in parallel. Our approach relies on Stokes theorem and simple Hankel-type matrix identities.

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