The Gaussian Moments Conjecture and the Jacobian Conjecture

We first propose what we call the Gaussian Moments Conjecture. We then show that the Jacobian Conjecture follows from the Gaussian Moments Conjecture. Note that the the Gaussian Moments Conjecture is a special case of [11, Conjecture 3.2]. The latter conjecture was referred to as the Moment Vanishing Conjecture in [7, Conjecture A] and the Integral Conjecture in [6, Conjecture 3.1] (for the one-dimensional case). We also give a counter-example to show that [11, Conjecture 3.2] fails in general for polynomials in more than two variables.