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.

The original paper proves GMC(1); it is compatible with higher-dimensional counterexamples.

Proved

Complex polynomials in one standard real Gaussian variable.

Source dated Jun 17, 2015 · Observed Oct 13, 2026

In this problem · publication v6

Special case · Proved

Relationship not independently reviewed. This concerns its use in this problem.

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.

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

Complex polynomials in one standard real Gaussian variable.

Source dated Jun 17, 2015 · Observed Oct 13, 2026 · Source passage →

Record details