Correlated Gaussian Moments Determine Polynomials up to Orthogonal Equivalence

Abstract

Let $P$ be a real polynomial of degree at most $m$ on ${\mathbb{R}}^d$. We construct finitely many correlated Gaussian moments that determine $P$ up to an orthogonal change of variables. The number of copies and the rational covariance matrices are specified in terms of $d,m$. On coefficient balls, the induced map on orthogonal orbits has a Hölder continuous inverse with an explicit exponent. For $m\ge 3$, the normalised local stationary-phase expansion at the critical branch through the origin equals the Gaussian moment series. Its coefficients recover the same finite moment vector, without assumptions on the other critical points. If the leading homogeneous part has an isolated complex critical point, we compute the twisted de Rham ranks and construct formal lattices on each fixed coupling support.