Proved
One-variable case proved ↗
Complex polynomials in one standard real Gaussian variable.
Source dated Jun 17, 2015 · Observed Oct 13, 2026
Theorem · version 1
The one-variable case
Statement
For P,Q ∈ ℂ[x] and a standard real Gaussian X, if E[P(X)^m] = 0 for all positive integers m, then E[Q(X)P(X)^m] = 0 for all sufficiently large m.
Assumptions
- One real Gaussian coordinate; complex coefficients.
- The eventual threshold may depend on Q.
Scope
The original paper proves GMC(1); it is compatible with higher-dimensional counterexamples.
Evidence
Source reports
Connections
In this problem · publication v5 · Selected context
Gaussian moments conjecture →
Special case · Proved
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v6
Gaussian moments conjecture →
Special case · Proved
Relationship not independently reviewed. This concerns its use in this problem.
Sources
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.
Proved
One-variable case proved ↗
Complex polynomials in one standard real Gaussian variable.
Source dated Jun 17, 2015 · Observed Oct 13, 2026 · Source passage →
Record details
- Claim ID
- claim-gaussian-moments-one-variable
- Statement version
- 1
- SHA-256
- da8cfccf49575db21b4e274b58f44390fd72214238c787231284d3030b1af6ed