The Gaussian moments conjecture and the Jacobian conjecture (Q2627999)

The authors propose the Gaussian Moments Conjecture \(\mathbf{GMC} (n)\), which says that if \(P \in \mathbb C [x_1, \dots, x_n]\) is a polynomial such that the moments \(\mathbb E (P(X)^m)=0\) for all \(m \geq 1\), where \(\mathbb E (X)\) is the expected value of a random variable \(X\), then for every \(Q \in \mathbb C [x_1, \dots, x_n]\), \(\mathbb E (P(X)^m Q(X))=0\) for \(m \gg 0\). This new conjecture is a special case of the Integral Conjecture of \textit{A. van den Essen} and \textit{W. Zhao} [J. Pure Appl. Algebra 217, No. 7, 1316--1324 (2013; Zbl 1279.13011)], for which they provide a counterexample. The main result in the paper is a proof that if \(\mathbf{GMC} (n)\) holds for all \(n \geq 1\), then the Jacobian conjecture also holds for all \(n \geq 1\).