Probability · Algebra · Conjecture
If every positive power of a complex polynomial has zero Gaussian expectation, must its products with any fixed polynomial eventually have zero expectation?
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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.
Precise statement
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.
Figures and explorable examples. Selecting a statement opens its exact wording.
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
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.
The all-dimensional assertion fails already at n = 3.
Conjecture · Primary formulation
DisprovedLong 2026Eventual mixed-moment vanishing in every dimension
The all-dimensional assertion fails already at n = 3.
Total degree four in three real coordinates.
Theorem · Counterexample
ProvedLong 2026A quartic witness in three real variables
Total degree four in three real coordinates. Ignoring extra coordinates gives failure for every n ≥ 3.
The original paper proves GMC(1); it is compatible with higher-dimensional counterexamples.
Theorem · Special case
ProvedDerksen et al. 2015The one-variable case
The original paper proves GMC(1); it is compatible with higher-dimensional counterexamples.
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.
Theorem · Conditional implication
ProvedDerksen et al. 2015The global implication to polynomial inversion
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.
Theorem · Conditional implication
ProvedLong 2026The same witness works in every higher dimension
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.
A different witness: degree three with four real coordinates.
Theorem · Supporting result
ProvedLong 2026A cubic witness in four real variables
A different witness: degree three with four real coordinates. The lower-dimensional quartic does not replace this degree statement.
This restriction is unresolved in Long’s dimension discussion.
Conjecture · Different statement · related variant
OpenThe remaining two-variable question
This restriction is unresolved in Long’s dimension discussion.
The identities hold for every positive integer order in the cited proof. A finite table of moments illustrates them; it does not prove the all-orders claim.
A quartic witness in three real variables →The identities hold for every positive integer order in the cited proof. A finite table of moments illustrates them; it does not prove the all-orders claim.
A cubic witness in four real variables →Assuming A quartic witness in three real variablesProved
The same witness works in every higher dimension
Under stated assumptionsIf 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.
Source-grounded explanation · independent explanation review: pending.
How the problem is attacked, and what each line of attack cannot yet reach.
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.
Supporting statements
Source-grounded explanation · independent explanation review: pending.
Long publishes the cubic four-variable and quartic three-variable constructions.
Publication date records the source; the paper describes discovery order without providing separate calendar dates.
Derksen, van den Essen and Zhao formulate the conjecture and prove its one-variable case.
Their implication to the Jacobian conjecture is global, rather than an equivalence in matching dimensions.
| Literature | Status OpenMath assessment · Oct 13, 2026.
|
|---|---|
| Formal | None linked Reported in sources: No formal proof project for these explicit Gaussian identities was located in the selected primary sources. Independent mathematical assessment is pending. |
| Computation | |
| Review | 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. |
Choose a starting point below. Each result links to its exact statement, cited passages and checks.
Consult the original question and its hypotheses.
Read the mathematical argument; inspect its statement for assessment and scope.
Harm Derksen · Arno van den Essen · Wenhua Zhao · Preprint · Jun 17, 2015
Conjecture 1.1, Theorem 1.6 and Proposition 4.2
Christopher D. Long · Preprint · Jul 20, 2026
Proposition 4.1, Theorem 5.1, Corollary 5.2, Section 7 and AI disclosure
Research direction
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
Small Counterexamples to the Gaussian Moments Conjecture · sources & artifacts →
Selected literature; source observations and coverage limits are retained in the publication details.
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.
Harm Derksen · Arno van den Essen · Wenhua Zhao
No specific AI contribution is attributed by this source record.
Christopher D. Long
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.
These are source credits; they do not imply OpenMath accounts, endorsements or independent mathematical review.