probability-statistics / Probability, Commutative Algebra
The Gaussian Moments Conjecture asks whether, for complex polynomials $P,Q$ in $n$ independent standard real Gaussian variables, $\mathbb{E}(P^m)=0$ for all $m\geq 1$ forces $\mathbb{E}(QP^m)=0$ for all large $m$. Explicit counterexamples with $\mathbb{E}(P^m)=0$ and $\mathbb{E}(QP^m)=m!\neq 0$ exist in three variables (a five-term quartic $P$) and four variables, so the conjecture is false in every dimension $n\geq 3$.
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Append-only history
Explicit counterexamples in dimensions 3 and 4, so GMC(n) fails for every n >= 3; GMC(1) was already known, and a separate human-authored preprint claims the remaining n = 2 case affirmatively
Research memory
The Gaussian Moments Conjecture asks whether, for complex polynomials $P,Q$ in $n$ independent standard real Gaussian variables, $\mathbb{E}(P^m)=0$ for all $m\geq 1$ forces $\mathbb{E}(QP^m)=0$ for all large $m$. Explicit counterexamples with $\mathbb{E}(P^m)=0$ and $\mathbb{E}(QP^m)=m!\neq 0$ exist in three variables (a five-term quartic $P$) and four variables, so the conjecture is false in every dimension $n\geq 3$.
Explicit counterexamples in dimensions 3 and 4, so GMC(n) fails for every n >= 3; GMC(1) was already known, and a separate human-authored preprint claims the remaining n = 2 case affirmatively
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