# The Gaussian moments conjecture

- kind: family
- status: refuted (evidence tier V0)
- year: 2017
- authors: H. Derksen, A. van den Essen and W. Zhao (arXiv 2015; Israel J. Math. 2017)
- in one line: GMC(n) — vanishing Gaussian moments of P force vanishing of Q·Pᵐ

**Statement.** Let $X=(X_1,\dots,X_n)$ be independent standard real Gaussians. $\mathrm{GMC}(n)$: if $P\in\mathbb C[x_1,\dots,x_n]$ has $\mathbb E[P(X)^m]=0$ for all $m\ge1$, then for every $Q$, $\mathbb E[Q(X)P(X)^m]=0$ for all large $m$.

The Gaussian moments conjecture is the probabilistic form of the Jacobian conjecture. It asks whether the kernel of Gaussian expectation is a Mathieu–Zhao subspace of the polynomial ring. Derksen, van den Essen and Zhao proved that [the Gaussian moments conjecture in every dimension implies the Jacobian conjecture](/wiki/Statement:jc-n.md), and that a non-invertible cubic-homogeneous Keller map in $r$ variables makes GMC($2r$) false.[1] The operators $\partial_i-x_i$ commute, have order one and constant leading coefficients, and by Gaussian integration by parts their images span $\ker\mathbb E$. So [GMC is the image conjecture for this one commuting family](/wiki/Statement:image-conjecture.md).[1]

| | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 | n = 6 |
|---|---|---|---|---|---|---|
| GMCₙ | proved | claimed true (V4; open) | refuted (witnessed) | refuted (witnessed) | refuted (witnessed) | refuted (witnessed) |
| JCₙ | proved | open | refuted (witnessed) | refuted (witnessed) | refuted (witnessed) | refuted (witnessed) |

## Status

- **Three variables and up are false**, with small explicit witnesses found independently of the Keller map. In three standard Gaussians, write $Z,W=(X_1\pm iX_2)/\sqrt2$ and $T=X_3$. Then $P=W+WZ-T^2-\tfrac32ZT^2-\tfrac12Z^2T^2$ and $Q=Z$ satisfy $\mathbb E[P^m]=0$ and $\mathbb E[QP^m]=m!$ for every $m\ge1$. Long also gives a six-term cubic in four variables.[2] His repository reports Lean 4 proofs `not_GMC_three` and `not_GMC_of_three_le` against an in-repository definition of GMC. They are pinned to Lean 4.34.0 and a fixed Mathlib commit, with an axiom audit.[3] The atlas has not rebuilt the project or audited the faithfulness of that definition. The tier V0 is the author's report of a kernel check.
- **One variable is true** (Derksen–van den Essen–Zhao, Proposition 4.2).
- **Two variables are claimed true.** Wilson's face-isolation proof is an unrefereed preprint, so the cell is drawn as a claim.[4] Long wrote of this case: “We do not settle this case.”[2]

If Wilson's proof holds, the threshold is sharp at three variables. The Jacobian conjecture's own threshold is also at three variables.

## Relations

- implies [jc-n](/wiki/Statement:jc-n.md) — Derksen, van den Essen and Zhao proved that the Gaussian moments conjecture in every dimension implies the Jacobian conjecture, and that a non-invertible cubic-homogeneous Keller map in $r$ variables makes GMC($2r$) false. The operators $\partial_i-x_i$ commute, have order one and constant leading coefficients, and by Gaussian integration by parts their images span $\ker\mathbb E$. [1]
- special case of [image-conjecture](/wiki/Statement:image-conjecture.md) — So GMC is the image conjecture for this one commuting family. [1]

## References

1. H. Derksen, A. van den Essen, W. Zhao, [The Gaussian Moments Conjecture and the Jacobian Conjecture](https://arxiv.org/abs/1506.05192), Israel J. Math. 219 (2017) 917–928.
2. C. D. Long, [Small Counterexamples to the Gaussian Moments Conjecture](https://arxiv.org/abs/2607.18186), arXiv:2607.18186 (20 July 2026).
3. [long-mathematics/gaussian-moments-counterexamples](https://github.com/long-mathematics/gaussian-moments-counterexamples), README and FORMALIZATION_STATUS.md.
4. M. Wilson, [A face-isolation proof of the two-variable Gaussian Moments Conjecture](https://arxiv.org/abs/2607.23887), arXiv:2607.23887 (26 July 2026). Unrefereed.

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