{"id":"openai-math/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables","pdf":"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","url":"https://github.com/openai/math/tree/main/preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026","tags":["affine-geometry","polynomial-rings","stable-coordinates"],"score":0,"title":"A stable coordinate that is not a coordinate in four variables","domain":"Algebraic and complex geometry","family":"049","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.AG","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-05","reactions":{},"formalized":true,"family_title":"A stable-coordinate counterexample in four variables"}