VibeMathedMath problems solved with AI
Sign in

The stable coordinate conjecture in four variables

A polynomial f∈C[x1,…,xn]f\in\mathbb C[x_1,\dots,x_n] is a coordinate if some polynomial automorphism sends it to x1x_1, and a one-stable coordinate if it is a coordinate of C[x1,…,xn,w]\mathbb C[x_1,\dots,x_n,w]. Shpilrain and Yu conjectured that a polynomial which becomes a coordinate after adjoining a variable was already a coordinate. Over C\mathbb C it holds in one, two and three variables (Shpilrain-Yu, using Abhyankar-Moh, cancellation for curves and surfaces, and Kaliman's theorem), so four variables is the first possible dimension for a counterexample. Is every one-stable coordinate in C[x1,x2,x3,x4]\mathbb C[x_1,x_2,x_3,x_4] a coordinate?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Affine algebraic geometry; polynomial automorphisms
Posed by
V. Shpilrain and J.-T. Yu, Affine varieties with equivalent cylinders, J. Algebra 251 (2002), Conjecture 3
Year posed
2002
Years open
24y
Solved
2026-10-05
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
22 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: with Q=x22−x42+x1x3Q=x_2^2-x_4^2+x_1x_3 and f=x1−2Q(Q(x2+x4)+x1x4)f=x_1-2Q(Q(x_2+x_4)+x_1x_4), a degree-five polynomial in C[x1,…,x4]\mathbb C[x_1,\dots,x_4], some automorphism of C[x1,…,x4,w]\mathbb C[x_1,\dots,x_4,w] sends ff to x1x_1 but none of C[x1,…,x4]\mathbb C[x_1,\dots,x_4] does. Since the conjecture holds in at most three variables, this is the smallest possible dimension, and one added variable suffices. Corollary 1.2: every fibre of ff is A3\mathbb A^3 and none is rectifiable. Not shown: anything about dimensions above four, which the release's earlier five-variable example already covered for five; the result is one explicit polynomial, not a classification.

What the AI did

Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems, published in the openai/math release (pinned commit adc7f12). The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. No Lean formalization accompanies it, and the README cautions that unformalized results could have issues. The construction adapts the method of the release's earlier cancellation manuscript.

Verification

No independent mathematician has checked this yet. Checked here: abstract, introduction, Theorem 1.1 and Corollary 1.2 of the TeX source, read against Shpilrain-Yu's Conjecture 3 as cited. The stabilisation is by explicit substitutions; non-coordinate status rests on a filtration, line-bundle lift and derivation-rigidity argument that was not refereed. The family's Lean formalization (AbhyankarSathaye) covers the 24 September companion's different polynomial, not this one, so this entry is unreviewed. The manuscript dates 5 October 2026, after the release's five-variable counterexample.

Sources

Changelog1 change

Discussion