A stable-coordinate counterexample in four variables
OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
OpenAI's result summary
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.
Read the manuscripts
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
MANUSCRIPT 01
A stable coordinate that is not a coordinate in four variables
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…
Full source 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.
This PDF is listed in OpenAI's formalization catalogue. See the scope and dependencies before interpreting coverage.
An explicit noncoordinate polynomial with affine three-space zero fibre
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.
This PDF is listed in OpenAI's formalization catalogue. See the scope and dependencies before interpreting coverage.
New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.