If every positive power of a complex polynomial has zero Gaussian expectation, must its products with any fixed polynomial eventually have zero expectation?
Long’s quartic witness refutes the universal statement in three variables, hence in every higher dimension. Dimension one is proved in the original source; dimension two remains a related open question.
For n independent standard real Gaussian variables X, let P and Q belong to ℂ[x₁,…,xₙ]. The hypothesis is E[P(X)^m] = 0 for every integer m ≥ 1. The proposed conclusion is: for each Q there is a threshold M, allowed to depend on Q, such that E[Q(X)P(X)^m] = 0 for every m ≥ M. Complex coefficients are essential; these are ordinary powers, not absolute moments.
Question paraphrased by OpenMath. Original wording is available in the sources.
Landscape
Figures and explorable examples. Selecting a statement opens its exact wording.
Check the witness
Exact Gaussian expectations: every pure moment E[P₃m] is 0, while the mixed moment E[Z P₃m] equals m! and never vanishes.
m
E[P₃^m]
E[Z P₃^m]
m!
1
0
1
1!
2
0
2
2!
3
0
6
3!
4
0
24
4!
5
0
120
5!
6
0
720
6!
7
0
5040
7!
8
0
40320
8!
Eight orders computed exactly; the identity for every m is the cited theorem. Nothing here bears on two variables, which remain open.
Visual setup
Zero pure moments do not force mixed moments to vanish
The three-variable witness has mixed moments m!, although every positive pure moment is zero.
Reading boundary. The table samples an exact identity from the paper. These four values alone do not establish the infinite sequence.
Select a statement for the polynomial, quantifiers and cited proof.
Text version
A table lists orders one through four: pure moments 0,0,0,0 and mixed moments 1,2,6,24. A dimension summary distinguishes the proved one-variable case, unresolved two-variable case and counterexamples from three onward.
Independent standard real Gaussian coordinates; complex polynomial coefficients. The displayed mixed moment uses Q₃ = Z.
Inspect the explicit polynomial and identity
●Proved
A quartic witness in three real variables
Total degree four in three real coordinates. Ignoring extra coordinates gives failure for every n ≥ 3.
Authored by OpenMath source curation on 2026-10-13 (AI-assisted curation). Independent review of this visual explanation is pending; linked assertions and relationships have their own assessments.
The all-dimensional assertion fails already at n = 3.
Long 2026
Conjecture · Primary formulation
✕DisprovedLong 2026
Eventual mixed-moment vanishing in every dimension
For every n ≥ 1 and every P∈C[x1,…,xn], if E[P(X)m]=0 for every m ≥ 1, then for each Q∈C[x1,…,xn] there is M such that E[Q(X)P(X)m]=0 for every m ≥ M.
Assumptions
X consists of n independent standard real Gaussians.
Positive integer powers; the threshold may depend on P and Q.
In this problem
The all-dimensional assertion fails already at n = 3.
This is a one-way implication. Refuting the Gaussian premise does not logically refute the Jacobian conclusion; the small Gaussian witnesses are not themselves Keller maps.
Derksen et al. 2015
Theorem · Conditional implication
●ProvedDerksen et al. 2015
The global implication to polynomial inversion
If the Gaussian Moments Conjecture holds in every positive dimension, then every polynomial map F: ℂⁿ → ℂⁿ in every n ≥ 1 with nonzero constant Jacobian determinant has a polynomial inverse.
Assumptions
The premise is the all-dimensional Gaussian conjecture, not one fixed dimension.
Complex polynomial maps with nonzero constant Jacobian determinant.
In this problem
This is a one-way implication. Refuting the Gaussian premise does not logically refute the Jacobian conclusion; the small Gaussian witnesses are not themselves Keller maps.
Additional independent coordinates do not change these expectations.
Long 2026
Theorem · Conditional implication
●ProvedLong 2026
The same witness works in every higher dimension
If complex polynomials P,Q in three independent standard real Gaussian variables satisfy E[P^m] = 0 and E[QP^m] = m! for every positive integer m, then for every n ≥ 3 these polynomials, independent of the extra coordinates, refute the Gaussian Moments Conjecture in n variables.
Assumptions
The exact three-variable identities hold for every positive integer m.
Additional coordinates are independent standard real Gaussians.
In this problem
Additional independent coordinates do not change these expectations. This extension leaves the two-variable case untouched and by itself gives no conclusion about the Jacobian conjecture.
This restriction is unresolved in Long’s dimension discussion.
Conjecture · Different statement · related variant
○Open
The remaining two-variable question
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.
In this problem
This restriction is unresolved in Long’s dimension discussion.
If complex polynomials P,Q in three independent standard real Gaussian variables satisfy E[P^m] = 0 and E[QP^m] = m! for every positive integer m, then for every n ≥ 3 these polynomials, independent of the extra coordinates, refute the Gaussian Moments Conjecture in n variables.
Limit Additional independent coordinates do not change these expectations. This extension leaves the two-variable case untouched and by itself gives no conclusion about the Jacobian conjecture.
The original all-dimensional Gaussian conjecture would imply the Jacobian conjecture. Failure of the former alone does not refute the latter, and this reduction is not dimension preserving.
Approaches
How the problem is attacked, and what each line of attack cannot yet reach.
Read the coefficient identity, not just sample moments
For either witness and its distinguished coordinate Z, the source derives E[A(Z)P^m] = m! [z^m] A(z)(1+z)^(m−1). Taking A = 1 or A = z gives the two headline identities.
Cannot yet reach A few moment values illustrate the mechanism; the coefficient identity establishes all positive orders.
Reported in sources: No formal proof project for these explicit Gaussian identities was located in the selected primary sources. Independent mathematical assessment is pending.
CComputation
None recorded
RReview
Not yet reviewed
OpenMath assessment: Long’s quartic witness refutes the universal statement in three variables, hence in every higher dimension. Dimension one is proved in the original source; dimension two remains a related open question.
Source comparison and reader curation; independent mathematical assessment remains pending.
OpenMath pilot collection · OpenMath source curation · assessed Oct 13, 2026 · review due Jan 13, 2027
Collection last checked Oct 13, 2026.
Sources
Choose a starting point below. Each result links to its exact statement, cited passages and checks.
Proposition 4.1, Theorem 5.1, Corollary 5.2, Section 7 and AI disclosure
Research activity · 1 entry
Contributions
Long is the human author. ChatGPT 5.6 Sol Pro and Claude Fable 5 receive distinct, self-reported witness-discovery credits; Derksen, van den Essen and Zhao supplied the original question and reduction.
Research direction
Explore the remaining dimension
The source leaves GMC(2) as the nearest unresolved restriction.
Higher-dimensional counterexamples cannot be projected down while preserving their moment identities.
Related statements · context, not additional proved progress