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)
Manuscripts (2)
A stable coordinate that is not a coordinate in four variables
October 5, 2026 · source · Lean formalization available · cite as \cite{OAI:A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026}
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.
An explicit noncoordinate polynomial with affine three-space zero fibre
September 24, 2026 · source · Lean formalization available · cite as \cite{OAI:An-explicit-noncoordinate-polynomial-with-affine-three-space-zero-fibre-September-24-2026}
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.
Result 049 · Algebraic and complex geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md