The Abhyankar-Sathaye embedding conjecture

A polynomial $f\in\mathbb C[z_1,\dots,z_n]$ is a coordinate if it is the first component of a polynomial automorphism of $\mathbb A^n$ . The Abhyankar-Sathaye (epimorphism, or embedding) conjecture asserts that if $\mathbb C[z_1,\dots,z_n]/(f)\cong\mathbb C^{[n-1]}$ then $f$ is a coordinate; geometrically, every hypersurface embedding $\mathbb A^{n-1}\hookrightarrow\mathbb A^n$ is rectifiable. For $n=2$ this is the Abhyankar-Moh-Suzuki theorem; Sathaye and Russell treated planes linear in one variable, and Kaliman-Venereau-Zaidenberg and Maubach handled four-variable equations $a(x)y+b(x,z,t)$ . Is every polynomial whose zero locus is isomorphic to affine $(n-1)$ -space a coordinate?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Affine algebraic geometry; polynomial automorphisms
Posed by: S. S. Abhyankar and A. Sathaye; the coordinate question is distinguished in A. Sathaye, On linear planes, Proc. Amer. Math. Soc. 56 (1976)
Year posed: 1976
Years open: 50y
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 45 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

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.

What the AI did

Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems, published in the openai/math release (pinned commit adc7f12). The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The main theorem has a Lean formalization in the release (Comparator challenge AbhyankarSathaye); a later family manuscript (5 October 2026) gives a stronger example with every fibre affine three-space.

Verification

No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against the conjecture as stated via Sathaye, Kraft and Blanc-van Santen. Lean-checked on the release's Comparator challenge AbhyankarSathaye (declaration OAI.AbhyankarSathaye.exists_noncoordinate_polynomial, listed in lean/formalization.yaml). Its statement was read here: for every $n\ge4$ there is $F\in\mathbb C[x_1,\dots,x_n]$ with a $\mathbb C$ -algebra isomorphism from the quotient by $(F)$ to a polynomial ring in $n-1$ variables, and no $\mathbb C$ -algebra automorphism sends a variable to $F$ . That is the headline, for all $n\ge4$ . Not rebuilt here. The paper states the three-variable case is not addressed. The explicit degree-five polynomial of the 5 October companion (all fibres affine) is unformalized.

The Abhyankar-Sathaye embedding conjecture

What was actually shown

What the AI did

Verification

Sources

Discussion