2026/07/20 by Christopher D. Long · 1 voice · 1 citation
Mathematics · #math.PR #math.AC #math.AG
We give explicit complex polynomials P,Q in three independent standard real Gaussian variables such that \mathbb E(Pm)=0, \mathbb E(QPm)=m!≠0 for every m≥1. In natural complex linear coordinates, P has five terms and total degree 4. Hence the Gaussian Moments Conjecture is false in every dimension n≥3. We also give a six-term cubic example in four variables, which was found first and already proves failure for every n≥4. Both examples follow from the same coefficient identity. The search was prompted by Levent Alpöge's public announcement of an explicit three-dimensional counterexample to the Jacobian Conjecture. Although the main theorem of Derksen, van den Essen, and Zhao is stated globally in dimension, its proof has fixed-dimensional content: a noninvertible cubic-homogeneous Keller map in r variables forces the failure of \mathrm GMC(2r). Tracking a standard Bass--Connell--Wright reduction of the announced map gives a conservative cubic-homogeneous counterexample in 79 variables, and hence a route-based failure of \mathrm GMC(158). That route is nonconstructive at the final Gaussian step and does not furnish explicit polynomials P,Q. The much smaller explicit failures in dimensions 4 and 3 below were not derived from the announced Jacobian map.