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Math-2445 · mathematics · Algebraic geometry

Equivalence of Affine-Space Embeddings Under Automorphisms

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Given embeddings $f,g:\mathbb C^d\hookrightarrow\mathbb C^n$, does there exist an automorphism $\Psi:\mathbb C^n\to\mathbb C^n$ such that $\Psi\circ f=g$?

Formulation references: [S1], [S2].

Short English statement

Given embeddings $f,g:\mathbb C^d\hookrightarrow\mathbb C^n$, does there exist an automorphism $\Psi:\mathbb C^n\to\mathbb C^n$ such that $\Psi\circ f=g$?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2445-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2445

Manuscript 1 (family 049, 2026-10-05): The explicit corollary identifies each hypersurface fiber with A3 and states that its embedding in A4 is not rectifiable. Comparing that closed embedding with the standard coordinate hyperplane disproves the proposed ambient-automorphism equivalence.

Manuscript 2 (family 049, 2026-09-24): The polynomial quotient is C in three variables but its defining polynomial is not an ambient coordinate. An automorphism taking that closed A3 embedding to the standard hyperplane would take its principal ideal to (x1), making the defining polynomial a coordinate up to a nonzero constant. This contradicts the main theorem and refutes the problem equivalence.

Equivalence of Affine-Space Embeddings Under Automorphisms

OpenAI's claim

Manuscript 1 (family 049, 2026-10-05): A degree-five polynomial in four complex variables is claimed to become a coordinate after adjoining one variable while failing to be a coordinate originally, despite every fiber being affine three-space. Manuscript 2 (family 049, 2026-09-24): An explicit polynomial in four complex variables is claimed to have affine three-space as its zero fiber without being a coordinate, with higher-dimensional examples obtained by adjoining variables.

Mathematical statement of the claimed result

Mathematical claim 1.
In $R=\mathbb C[x_1,x_2,x_3,x_4]$, set \[ Q=x_2^2-x_4^2+x_1x_3,\qquad f=x_1-2Q\left(Q(x_2+x_4)+x_1x_4\right). \] OpenAI claims a complex-algebra automorphism of $R[w]$ sending $f$ to $x_1$, and no such automorphism of $R$. It further claims that every hypersurface $f^{-1}(\lambda)$, $\lambda\in\mathbb C$, is isomorphic to affine three-space but its embedding in affine four-space cannot be straightened to a coordinate hyperplane by a polynomial automorphism.
Mathematical claim 2.
In $R=\mathbb C[h,u,v,w]$, define \[ x=u^3+hv,\quad y=-u^2+hw,\quad s=2u^3v+3u^4w+h(v^2-3u^2w^2)+h^2w^3, \] so $x^2+y^3=hs$. Set $p=-2s^2x+3sy^2-3s^3y$ and $F=h-p-1$. OpenAI claims \[ R/(F)\simeq\mathbb C^{[3]},\qquad\nabla F(2,0,-1/2,1/2)=0. \] Thus the zero fiber is affine three-space but $F$ is not a polynomial coordinate of $R$, since a coordinate has nowhere-vanishing gradient.

Outline of the argument

Manuscript 1 (family 049, 2026-10-05): Two hypersurface presentations and determinant-preserving matrix changes identify the polynomial ring and the distinguished parameter. An exponential of a locally nilpotent derivation gives the stabilized coordinates, while a parameter filtration and the rigidity of its associated bundle obstruct an unstabilized coordinate system.

Manuscript 2 (family 049, 2026-09-24): The construction perturbs the two entries of a cusp parametrization by independent parameter multiples and uses their divisibility relation. Explicit translations and inverse algebra maps identify the zero-fiber ring as a polynomial algebra; the separate ambient obstruction prevents rectifying its embedding.

Formal verification

For family 049, the scope page documents the following selections. The selected statement supplies noncoordinate polynomials with affine-space quotients in all ambient dimensions at least four. The companion derivation statement gives commuting independent locally nilpotent derivations with the specified common kernel; ambient dimension three is excluded. Manuscript 1 is not among the papers selected by that page, so the family link does not formalize its main conclusion. It does not establish formal coverage of the complete problem clause. The documented statement covers the main claim of manuscript 2. Its selected declaration also covers the clause used for this problem. Comparator files: AbhyankarSathaye.lean, CommutingDerivations.lean. Declarations: OAI.AbhyankarSathaye.exists_noncoordinate_polynomial, OAI.AbhyankarSathaye.CommutingDerivations.commuting_derivations_with_ordinary_extras.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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