Result 049, Algebraic and complex geometry

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.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

An extra variable can make a polynomial into a basic coordinate even when no polynomial change of coordinates can do so beforehand. The reported example distinguishes a space's intrinsic simplicity from how simply it sits inside another space.

What changes?

The manuscript reports an explicit degree-five polynomial in four complex variables that is not a coordinate: it cannot become one of the variables through an invertible polynomial change of variables. After adding one variable, it can. Each fiber, obtained by fixing the polynomial's value, is isomorphic to affine three-space, the space of three complex coordinates. Yet no such fiber can be moved to a coordinate hyperplane by an invertible polynomial transformation of the surrounding four-dimensional space.

What does that help mathematicians do?

The claimed example contradicts both the Stable Coordinate conjecture in four variables and the Abhyankar–Sathaye embedding conjecture in ambient dimension four. Researchers could therefore no longer treat either becoming a coordinate after adding a variable or having every fiber be affine space as sufficient to guarantee an original coordinate. The companion abstract also reports embedding counterexamples in every ambient dimension at least four by adjoining variables.

Are there practical applications?

The immediate value is foundational: the example separates two questions that can look deceptively similar, recognizing an affine space and straightening its embedding by polynomial transformations. It also identifies a limit of adding variables as a simplification technique. Any criterion for recovering an original coordinate would need assumptions that exclude this reported example.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

A stable coordinate that is not a coordinate in four variables

October 5, 2026 14 pages

We construct an explicit degree-five polynomial over C\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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

An explicit noncoordinate polynomial with affine three-space zero fibre

September 24, 2026 6 pages Main result formalized in Lean

We construct an explicit counterexample to the Abhyankar–Sathaye conjecture: a noncoordinate polynomial in four complex variables whose zero fibre is affine three-space. Adjoining variables gives counterexamples in every ambient dimension at least four.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:An-explicit-noncoordinate-polynomial-with-affine-three-space-zero-fibre-September-24-2026,
  author = {{OpenAI}},
  title = {{An explicit noncoordinate polynomial with affine three-space zero fibre}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-explicit-noncoordinate-polynomial-with-affine-three-space-zero-fibre-September-24-2026/paper.pdf}{OAI:An-explicit-noncoordinate-polynomial-with-affine-three-space-zero-fibre-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/049.md.

A stable-coordinate counterexample in four variables

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The Abhyankar–Sathaye conjecture predicts that a polynomial defining an affine-space quotient must be an ambient coordinate. For every n≥4n\ge4, the formalized counterexample gives F∈C[x1,…,xn]F\in\mathbb C[x_1,\ldots,x_n] with quotient C[x1,…,xn]/(F)≅C[y1,…,yn−1]\mathbb C[x_1,\ldots,x_n]/(F)\cong\mathbb C[y_1,\ldots,y_{n-1}], although FF is not a coordinate.

A companion gives n−1n-1 commuting, locally nilpotent derivations, linearly independent over the polynomial ring, whose common kernel is generated by the construction's explicit polynomial and contains no ambient coordinate. The extra n−4n-4 derivations are ordinary partial derivatives in the added variables. The three-variable case is not covered.

Comparator links

Result Comparator statement
Noncoordinate polynomial in every dimension at least four AbhyankarSathaye.lean
Commuting locally nilpotent derivations CommutingDerivations.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.