A face-isolation proof of the two-variable Gaussian Moments Conjecture

Abstract

Let $(X,Y)$ be independent standard real Gaussian random variables, and let $(P\in\mathbb C[X,Y])$ satisfy $\mathbb E!\left(P(X,Y)^m\right)=0$ for every $(m\ge1)$. Using the complex coordinates $Z=\frac{X+iY}{\sqrt2}, \qquad W=\frac{X-iY}{\sqrt2},$ we prove that the monomial support of $(P)$ is strictly one-sided with respect to the weight $\operatorname{wt}(Z^aW^b)=a-b$. Thus either every monomial occurring in $(P)$ has positive weight, or every monomial has negative weight. It follows that, for every $(Q\in\mathbb C[X,Y])$, $\mathbb E!\left(Q(X,Y)P(X,Y)^m\right)=0$ whenever $(m>\deg Q)$. This proves the two-variable Gaussian Moments Conjecture with the explicit threshold $m\ge\deg Q+1$. The main ingredient is a prime-isolation theorem for exposed faces of the Newton polygon of $(P)$. A $(p)$-adic valuation argument at moment indices of the form $(m=qp)$ separates the contribution of a chosen face from all remaining multinomial strata. Frobenius reduction then forces the constant terms of all positive powers of the associated one-variable Laurent polynomial to vanish. The theorem of Duistermaat and van der Kallen implies that the face has weights of one strict sign, while a planar convex-geometric argument rules out support containing weights of both signs. Combined with the known one-variable case and counterexamples in dimensions $(n\ge3)$, this determines the dimensions in which the Gaussian Moments Conjecture holds.