Open
Two-variable case remains open in the cited survey ↗
The three-variable counterexample does not settle dimension two.
Source dated Jul 20, 2026 · Observed Oct 13, 2026
Conjecture · version 1
The remaining two-variable question
Statement
For P,Q ∈ ℂ[x₁,x₂] and independent standard real Gaussians X₁,X₂, does E[P(X)^m] = 0 for every positive integer m imply E[Q(X)P(X)^m] = 0 for all sufficiently large m?
Assumptions
- Exactly two real Gaussian coordinates.
- Complex coefficients and a Q-dependent threshold.
Scope
This restriction is unresolved in Long’s dimension discussion.
Evidence
Source reports
Connections
In this problem · publication v6 · Selected context
Gaussian moments conjecture →
Different statement · related variant · Open
Relationship not independently reviewed. This concerns its use in this problem.
Sources
Small Counterexamples to the Gaussian Moments Conjecture →
- Proposition 4.1, Theorem 5.1, Corollary 5.2, Section 7 and AI disclosure → · Original passage ↗
Recorded wording
Gives a cubic four-variable witness and a quartic three-variable witness, both with zero pure moments and nonzero mixed moments of every order.
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.
Christopher D. Long · Preprint · Publication date Jul 20, 2026
Revision & attribution
Source revision: 2026-07-20. Observed Oct 13, 2026.
Long credits ChatGPT 5.6 Sol Pro with the four-variable polynomial, Claude Fable 5 with the three-variable reduction and checks, and ChatGPT with editing. Long is the paper’s author and responsible for its mathematical claims.
The Gaussian Moments Conjecture and the Jacobian Conjecture →
- Conjecture 1.1, Theorem 1.6 and Proposition 4.2 → · Original passage ↗
Recorded wording
Introduces Gaussian moment vanishing, proves the one-variable case and gives a global implication to the Jacobian conjecture.
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.
Harm Derksen · Arno van den Essen · Wenhua Zhao · Preprint · Publication date Jun 17, 2015
Revision & attribution
Source revision: 2015-06-17. Observed Oct 13, 2026.
No specific AI contribution is attributed by this source record.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Open
Two-variable case remains open in the cited survey ↗
The three-variable counterexample does not settle dimension two.
Source dated Jul 20, 2026 · Observed Oct 13, 2026 · Source passage →
Record details
- Claim ID
- claim-gaussian-moments-two-variable
- Statement version
- 1
- SHA-256
- 9f01744f9e66b2f816a765c3873df50d921931443f6572a6bedaca367a235b67