Abstract
Let G be a bounded open subset of Euclidean space with real algebraic boundary. In a first part of the talk we consider the case where $G={x: g(x) <=1}$ for some quasi-homogeneous polynomial "g" and derive several properties of "G" as well as the non-Gaussian integral \int exp(-g)dx. In particular we show that the volume of G is a convex function of the coefficients of "g" and solve a generalized version of the Lowner-John problem.
Next, we consider a more general case and under the assumptions that the degree "d" of the algebraic boundary is known 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 the boundary. 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. Our approach relies on Stokes theorem and simple Hankel-type matrix identities.