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Math-2329 · mathematics · Algebraic geometry

Stable Coordinate Conjecture for Complex Polynomials

Status Partially solved · unsolvedmath.comLLM status Open unverified

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.
If a polynomial $P\in\mathbb{C}[x_1,\ldots,x_n]$ becomes equivalent to $x_1$ after adjoining finitely many variables, must $P$ already be a coordinate?

Formulation references: [S1], [S2].

Short English statement

If a polynomial $P\in\mathbb{C}[x_1,\ldots,x_n]$ becomes equivalent to $x_1$ after adjoining finitely many variables, must $P$ already be a coordinate?

Sources

Catalogue entry

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

Contributions 1

Math-2329-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2329

Manuscript 1 (family 047, 2026-09-23): The primary manuscript's explicit stable-coordinate corollary gives a five-variable complex polynomial which becomes a coordinate after one new variable but was not a coordinate before it. That single instance disproves the problem all-n stable-coordinate implication.

Manuscript 2 (family 049, 2026-10-05): The main theorem gives a polynomial in four complex variables which an automorphism sends to x1 after adjoining one variable, but no original-ring automorphism does so. This is precisely a stable-coordinate counterexample.

Stable Coordinate Conjecture for Complex Polynomials

OpenAI's claim

Manuscript 1 (family 047, 2026-09-23): An explicit complex affine fourfold is claimed to become affine five-space after one stabilization while itself failing to be affine four-space, with stable-coordinate and nontrivial affine-fibration consequences. Manuscript 2 (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.

Mathematical statement of the claimed result

Mathematical claim 1.
In $P=\mathbb C[p,s,u,F,J]$, define \[ x=s^2+u^3+p^2F,\qquad H=x^2F-(1+2sx)J-p^2J^2-pu,\qquad A=P/(H). \] OpenAI claims that $A$ is integral of dimension four and for an independent variable $w$, \[ A[w]\simeq\mathbb C^{[5]},\qquad A\not\simeq\mathbb C^{[4]}. \] The notation $\mathbb C^{[j]}$ denotes a polynomial algebra in $j$ variables. It also claims that $H$ is a coordinate of $P[w]$ but not of $P$, where coordinate means an element belonging to a polynomial coordinate system. These assertions give one-variable cancellation and stable-coordinate counterexamples in the indicated dimensions. Writing $X=\operatorname{Spec}A$, the affine-fibration corollary claims that \[ \pi:X\to\mathbb A^1,\quad x\mapsto p(x),\qquad \rho:\mathbb A^5\to\mathbb A^2,\quad z\mapsto(p(z),H(z)) \] are smooth and surjective with relative dimension three; every fiber over a base point $y$ is $\mathbb A^3_{\kappa(y)}$, yet neither map is locally a product with $\mathbb A^3$ in the Zariski topology. Thus the examples satisfy faithful flatness and geometric integrality of the fibers.
Mathematical claim 2.
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.

Outline of the argument

Manuscript 1 (family 047, 2026-09-23): A locally nilpotent derivation and an explicit inverse polynomial change give the cylinder isomorphism. To obstruct cancellation, a valuation along the exceptional parameter computes the associated graded affine modification; the resulting rigidity distinguishes the original coordinate ring from a polynomial algebra.

Manuscript 2 (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.

Formal verification

For family 047, the scope page documents the following selections. The selected construction is a four-dimensional complex domain whose one-variable polynomial extension is a five-variable polynomial ring, while the original ring is not polynomial. The other affine-fibration and stable-coordinate consequences are separate. The documented statement covers the main claim of manuscript 1. It does not establish formal coverage of the complete problem clause. Comparator files: ComplexCancellation.lean. Declarations: OAI.ComplexCancellation.main.

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 2 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.

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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