The Abhyankar-Sathaye embedding conjecture

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An explicit noncoordinate polynomial with affine three-space zero fibre (companion) evidence for this event

Sathaye 1976, On linear planes evidence for this event

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Lean Comparator challenge CommutingDerivations (OpenAI math release) evidence for this event

VibeMathed record: The Abhyankar-Sathaye embedding conjecture evidence for this event

Lean Comparator challenge AbhyankarSathaye (OpenAI math release) evidence for this event

Theorem 1.1: an explicit $F\in\mathbb C[h,u,v,w]$, built from a perturbed cusp parametrisation, has $\mathbb C[h,u,v,w]/(F)\cong\mathbb C^{[3]}$ but a critical point on the fibre $F=-1$, so $F$ is not a coordinate; adjoining variables gives counterexamples for every $n\ge4$. The 5 October companion gives a degree-five $f$ in four variables with every fibre isomorphic to $\mathbb A^3$ and no fibre rectifiable, so the conjecture fails even for polynomials with all fibres affine. Not shown: the case $n=3$ (whether every embedded affine plane in $\mathbb A^3$ is rectifiable), which stays open. parent of this event

Lean Comparator challenge StableCoordinateFour (OpenAI math release) evidence for this event

The stable coordinate conjecture in four variables evidence for this event

The Abhyankar-Sathaye embedding conjecture parent of this event

Kraft, Challenging problems on affine n-space, Seminaire Bourbaki 802 (1996) evidence for this event