Abhyankar–Sathaye coordinate conjecture in four variables
Follow the latest progress by email.
The Abhyankar–Sathaye conjecture asks whether a polynomial whose zero fibre is an affine space must be a coordinate polynomial. Here the first unresolved ambient case is a polynomial in four complex variables with zero fibre isomorphic to affine three-space.
Why it matters and what counts as progress
Why it matters: The four-variable case tests whether an abstract affine-space hypersurface must also be a coordinate hyperplane in its ambient space.
What counts as progress: Prove the four-variable implication, or construct a noncoordinate polynomial with the required affine-space zero fibre.
How the problem developed
- 1976 · Open problem [1]
Sathaye Asked whether a polynomial in four variables with zero fibre isomorphic to affine three-space must have all fibres isomorphic to affine three-space.
- 1994 · Conjecture [1]
Hanspeter Kraft Surveyed the Abhyankar–Sathaye conjecture that every hypersurface embedding of affine three-space in affine four-space is defined by a coordinate polynomial.
- 24 Sept 2026 · Reported claim [2]
OpenAI The preprint reports a noncoordinate polynomial whose zero fibre is affine three-space, a counterexample to the stated embedding conjecture if correct.
AI contribution
An explicit noncoordinate polynomial with affine three-space zero fibre
An explicit polynomial F in four complex variables has zero-fibre coordinate ring isomorphic to C[3], while its gradient vanishes at a point, so F is not a coordinate. Adding variables extends this counterexample to every ambient dimension at least four.
Sources
Website timeline
Publication dates and editorial changes.
OpenAI reports a four-variable Abhyankar–Sathaye counterexample