  Let $G$ be a bounded open subset of Euclidean space with real algebraic
boundary $\Gamma$. Under the assumption that the degree $d$ of $\Gamma$ 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 $\Gamma$. 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.
