The Gaussian Measure On Algebraic Varieties

Abstract

We prove that the ring $\Aff{\R}{M}$ of all polynomials defined on a real algebraic variety $M\subset\R^n$ is dense in the Hilbert space $L^2(M,e^{-|x|^2}\deμ)$, where $\deμ$ denotes the volume form of $M$ and $\deν=e^{-|x|^2}\deμ$ the Gaussian measure on $M$.