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A stable coordinate that is not a coordinate in four variables
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Abstract
We construct an explicit degree-five polynomial over $\mathbf C$ that is not a coordinate in four variables but becomes one after adjoining a single variable. This gives a counterexample to the stable coordinate conjecture in four variables. Every fiber is isomorphic to affine three-space, yet its embedding in affine four-space is not rectifiable. Thus the example also disproves the Abhyankar–Sathaye embedding conjecture in ambient dimension four.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope note 049: A stable-coordinate counterexample in four variables (release Lean docs, October 2026). Describes the formalized scope: a degree-five one-stable noncoordinate in four variables whose fibres are all affine three-space and nonrectifiable, the Abhyankar–Sathaye statement for every n ≥ 4, and a related commuting-derivations statement, with links to the three Comparator challenges. [family-level]
- OpenAI math release, StableCoordinateFour.lean (comparator statement) (openai/math, added in the 7 October 2026 update). Lean statement
OAI.StableCoordinate.principal_packageencoding Theorem 1.1 (degree five, one-stable coordinate, not a coordinate) and Corollary 1.2 (every fibre ≅ A³, not rectifiable); the proof field in this statement file issorry. - OpenAI math release, AbhyankarSathaye.lean (comparator statement) (openai/math, October 2026). Lean statement
OAI.AbhyankarSathaye.exists_noncoordinate_polynomial(for every n ≥ 4 a noncoordinate F with quotient ≅ ℂ^[n-1]), the Abhyankar–Sathaye consequence shared by both papers; the proof field in this statement file issorry. [family-level]
Related prior literature mentioned by others
- Vladimir Shpilrain and Jie-Tai Yu, Affine varieties with equivalent cylinders (arXiv:math/0110324, 2001). The AI Proved What? episode links this paper for the stable coordinate conjecture and its low-dimensional background; it is also in the October manuscript's bibliography.
Possibly relevant literature (found by the agent, not checked)
- Vladimir L. Popov, Around the Abhyankar–Sathaye conjecture (arXiv:1409.6330, 2014). Survey-type paper on the conjecture and on commuting locally nilpotent derivations, relevant to the paper's claim that Corollary 1.2 disproves the Abhyankar–Sathaye conjecture in ambient dimension four.
- Shulim Kaliman and Mikhail Zaidenberg, Venereau polynomials and related fiber bundles (arXiv:math/0308191, 2003). Studies Vénéreau polynomials on A⁴ with all fibres ≅ A³ via fibre bundles, the testing ground the introduction discusses alongside Lewis and Blanc–Poloni.
- Parnashree Ghosh and Neena Gupta, On the triviality of a family of linear hyperplanes (arXiv:2206.15210, 2022/2023). Gives higher-dimensional families of linear hyperplanes for which the Abhyankar–Sathaye conjecture holds, a point of comparison for the paper's four-dimensional counterexample.
Digests and expositions
- AI Proved What?, One variable too few? A stable coordinate that is not a coordinate (episode 049, October 2026). This AI-assisted video and transcript explain both constructions, outline the October paper's filtration and derivation obstruction, and distinguish it from the September paper's critical-point argument. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 049
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026,
author = {{OpenAI}},
title = {{A stable coordinate that is not a coordinate in four variables}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/stable-coordinate-four-variables.pdf}{OAI:A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/049.md. The Lean statements have not been checked by people against the paper's claims.
Family 049: A stable-coordinate counterexample in four variables
Constructs a polynomial in four complex variables that is not a coordinate but becomes one after adjoining a single variable, disproving the Stable Coordinate conjecture in four variables. Every fiber is affine three-space, yet none of its embeddings is rectifiable, also disproving the Abhyankar–Sathaye conjecture even when all fibers are affine spaces.
Other papers in this family:
- An explicit noncoordinate polynomial with affine three-space zero fibre · September 24, 2026