Family 049 — Affine space embedded in a way coordinates cannot straighten

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Family 049 · Algebraic and complex geometry
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-12

1. The question

A copy of affine three-space can be described by a complicated equation inside affine four-space. Does its complicated appearance always disappear after a polynomial change of ambient coordinates?

The Abhyankar–Sathaye embedding conjecture predicts that a hypersurface isomorphic to affine space can be straightened into a coordinate hyperplane in characteristic zero. It concerns the embedding, not merely the abstract shape of the hypersurface.

Family 049 contains two manuscripts claiming explicit counterexamples in four variables. They demonstrate different phenomena. M1 gives a polynomial that is not a coordinate, becomes one after adding a single variable, and has every fiber isomorphic to affine three-space. M2 gives a different polynomial whose zero fiber is affine three-space but which has a critical point on another fiber.

The critical point makes M2’s obstruction easy to state and preserves it after adjoining any number of unused variables. M1 cannot use that obstruction: its stable-coordinate property requires a subtler argument involving additive group actions.

2. Background and essential definitions

Coordinates are global polynomial data

A polynomial $f\in\mathbb C[x_1,\ldots,x_n]$ is a coordinate if other polynomials complete it to a coordinate system:

$$\mathbb C[x_1,\ldots,x_n] =\mathbb C[f,f_2,\ldots,f_n].$$

Equivalently, a polynomial automorphism carries $f$ to the first standard coordinate. An automorphism requires a polynomial inverse, not just a locally invertible or analytic change of variables.

For example, $x+y^2$ is a coordinate in two variables because the change $(x,y)\mapsto(x+y^2,y)$ is inverted by $(u,v)\mapsto(u-v^2,v)$. Every fiber is a parabola, and the same ambient automorphism straightens all of them at once.

A polynomial is one-stably a coordinate if it becomes a coordinate after adjoining one new independent variable. The Stable Coordinate Conjecture asks whether this can happen without the polynomial already being a coordinate.

An abstract fiber and its embedding

The fiber $f^{-1}(\lambda)$ has coordinate ring

$$\mathbb C[x_1,\ldots,x_n]/(f-\lambda).$$

Saying this ring is polynomial in $n-1$ variables describes the fiber intrinsically. An isomorphism may use formulas that work only after the equation $f=\lambda$ has been imposed.

The embedding is rectifiable if a polynomial automorphism of the entire ambient affine space carries the fiber to a coordinate hyperplane. Extending coordinates off the fiber is the extra requirement. Thus even a full polynomial parametrization of the hypersurface need not solve the embedding problem.

The Abhyankar–Sathaye conjecture asks whether

$$\mathbb C[x_1,\ldots,x_n]/(f)\simeq\mathbb C[t_1,\ldots,t_{n-1}] \quad\Longrightarrow\quad f\text{ is a coordinate}.$$

Critical points give one obstruction

A critical point of $f$ is a point where all partial derivatives vanish. A coordinate polynomial has no critical points: it is a row of the Jacobian matrix of a polynomial automorphism, and that matrix is invertible everywhere.

This gives a useful asymmetry. A critical point anywhere in the ambient space disproves coordinateness, even when it lies far from the zero fiber. The zero fiber can still be perfectly smooth and abstractly affine space.

This obstruction is unavailable when the polynomial becomes a coordinate after adding unused variables, since its zero derivatives in those new directions would leave any old critical point intact.

3. Development history and previous partial results

The two-variable embedding problem is solved affirmatively by the Abhyankar–Moh theorem and Suzuki’s related work: an affine line embedded as such a plane curve can be straightened.

In three variables, important positive results impose conditions on the equation or its general fiber. Sathaye treated planes defined by polynomials linear in one variable, and Russell extended that result to arbitrary fields. Kaliman proved that a polynomial on $\mathbb C^3$ with general fiber $\mathbb C^2$ is a coordinate. This does not resolve the full three-variable embedding question from a single affine-plane fiber.

Together with curve and surface cancellation, these results imply the Stable Coordinate Conjecture over $\mathbb C$ in at most three original variables, as explained by Shpilrain–Yu. Four variables are therefore the first possible dimension for M1’s stable-coordinate counterexample.

The Vénéreau polynomials have been important candidates and testing grounds in four variables. Lewis proved that the second Vénéreau polynomial is a coordinate. Blanc–Poloni developed affine-bundle and bivariable methods, revisited that result, and established one-stability for a wider family. Exhibiting stable coordinates was thus already possible; ruling out every original coordinate system remained the separate challenge.

For the embedding problem in four variables, Kaliman–Vénéreau–Zaidenberg and Maubach proved affirmative results for equations of the form $a(x)y+b(x,z,t)$, with the coefficient depending on only one variable. More recent work of Ghosh, Gupta, and Pal treats related linear hypersurfaces under further hypotheses. M2 is designed outside those restricted equation classes.

Shpilrain–Yu used ambient critical loci to distinguish inequivalent embeddings and asked whether an affine-space zero fiber forces the gradient to be nowhere zero. M2 claims a negative answer, while M1 claims an obstruction even with every fiber affine space.

4. What the new manuscripts claim

M1: degree five, all fibers affine, one stabilization sufficient

In four independent variables, put

$$Q=x_2^2-x_4^2+x_1x_3, \qquad f=x_1-2Q\bigl(Q(x_2+x_4)+x_1x_4\bigr).$$

Theorem 1.1 of M1 claims that this degree-five polynomial is not a coordinate of $\mathbb C[x_1,x_2,x_3,x_4]$, but becomes a coordinate after adjoining one variable $w$.

Corollary 1.2 states that for every $\lambda\in\mathbb C$,

$$f^{-1}(\lambda)\simeq\mathbb A^3,$$

and that none of these embeddings in $\mathbb A^4$ is rectifiable. If any one fiber could be taken to a coordinate hyperplane, its prime principal ideal would be taken to the ideal of a coordinate. Its defining polynomial would differ from that coordinate by a nonzero constant factor, contradicting the asserted noncoordinateness of $f$.

This simultaneously claims failure of stable coordinates in the smallest possible ambient dimension and failure of the embedding conjecture there.

M2: an affine zero fiber with a critical point elsewhere

Use four different independent variables $h,u,v,w$, and define

$$x=u^3+hv,\qquad y=-u^2+hw,$$

$$s=2u^3v+3u^4w+h(v^2-3u^2w^2)+h^2w^3.$$

These abbreviations satisfy $x^2+y^3=hs$. Set

$$p=-2s^2x+3sy^2-3s^3y, \qquad F=h-p-1.$$

Theorem 1.1 of M2 claims

$$\mathbb C[h,u,v,w]/(F)\simeq\mathbb C[t_1,t_2,t_3], \qquad \nabla F(2,0,-1/2,1/2)=0.$$

The stated critical point lies on $F=-1$, not on the smooth zero fiber. Hence $F$ is not a coordinate despite its affine-three-space zero fiber.

Adding independent variables preserves both the polynomial zero-fiber quotient and the critical point. M2 therefore claims embedding counterexamples in every ambient dimension at least four. In contrast, M1’s particular obstruction disappears after one added variable by design.

5. Why the advance matters

Importance: 4/5. If correct, these examples would settle major affine-embedding and stable-coordinate questions negatively, with explicit formulas in the first possible dimension for stable-coordinate failure.

They expose three distinct levels of information: one fiber is affine space; every fiber is affine space; the polynomial is a coordinate. The claimed examples separate all the implications needed for the corresponding conjectures.

M2 would show that the intrinsic simplicity of one fiber says little about critical behavior elsewhere. M1 goes further in a different direction: even uniform affine-space fibers would fail to provide a single ambient coordinate system. Stabilization would erase that obstruction.

Neither manuscript needs an exotic ambient variety. The ambient space is ordinary $\mathbb A^4$; the complexity lies in how a distinguished polynomial organizes it.

6. The main idea

M1: coordinates on the ring need not include the chosen element

M1 presents a four-variable polynomial ring as a hypersurface in five variables, with a distinguished element $p$. A determinant-preserving matrix substitution makes the defining equation linear in a new variable. Eliminating that variable identifies the quotient with a polynomial ring and turns $p$ into the displayed degree-five $f$.

A second explicit substitution, using the exponential of a locally nilpotent derivation, makes $p$ part of a coordinate system after one extra variable. A locally nilpotent derivation acts like differentiation along an algebraic additive-group action; its exponential terminates on every polynomial.

To obstruct coordinateness before stabilization, the manuscript places an integer-indexed filtration on the ring, including negative degrees. If $p$ were a coordinate, differentiating in another coordinate would induce a prohibited additive action on a simpler graded algebra associated with an affine quadric.

A line-bundle lift makes a key ideal principal. The generator has a highest part factoring as $(au-d)(au+d)$. The resulting invariance constraints force a coefficient to be both a polynomial in variables of nonnegative weight and an element of negative weight. M1 presents this incompatibility as the final obstruction. It adapts the method of family 047, but here the original ring itself is already polynomial.

M2: use a cusp relation to build the affine fiber

M2 starts with the cusp parametrization $(u^3,-u^2)$, for which $x^2+y^3=0$, and perturbs both entries by multiples of $h$. The quotient of $x^2+y^3$ by $h$ supplies $s$.

On the zero fiber, the translation

$$X=x+s^3,\qquad Y=y-s^2$$

turns an auxiliary relation into $s=X^2+Y^3$. This identifies an auxiliary algebra with a polynomial ring in two variables. A lifting lemma supplies the third variable by solving compatible linear equations whose coefficients generate the unit ideal.

That unit-ideal condition is important: it allows polynomial formulas valid everywhere, without dividing by a coefficient that could vanish. The appendix records coordinates and their polynomial inverse for the entire zero fiber.

The noncoordinate conclusion then comes from the separately stated critical point in the original ambient four-space. The construction distinguishes a global parametrization of the zero fiber from a global automorphism of the ambient space.

7. Scope, limitations, and open questions

Both constructions are over $\mathbb C$. M1 concerns a one-stable coordinate in four original variables, and M2 concerns an affine zero fiber with a persistent critical-point obstruction. M2 is not a stable-coordinate example; adjoining unused variables cannot remove its critical point.

The family does not give an affine-space cancellation counterexample in dimension three. In M1 the ambient fourfold and every fiber are already affine spaces; what allegedly fails is rectification of the chosen polynomial. Cancellation and coordinate questions have relationships, but their negations cannot be exchanged without an argument.

The three-variable Abhyankar–Sathaye problem from a single affine-plane fiber is not addressed. The earlier affirmative special cases remain distinct from the four-variable counterexamples claimed here.

Further questions include identifying additional hypotheses on an affine-space fibration that force a coordinate, and understanding how stabilization changes the possible polynomial embeddings.

8. Sources and further reading

M1’s Theorem 1.1 gives the degree-five stable-coordinate claim and Corollary 1.2 its all-fibers consequence. M2’s Theorem 1.1 gives the zero-fiber isomorphism and critical point, while its appendix provides explicit inverse coordinate maps.

The primary works linked above trace the low-dimensional embedding theorems, Vénéreau examples, and critical-locus approach. The useful distinction throughout is between coordinates on a hypersurface and coordinates on its entire ambient affine space.

Source manuscripts

• M1. OpenAI, A stable coordinate that is not a coordinate in four variables (October 5, 2026).
• M2. OpenAI, An explicit noncoordinate polynomial with affine three-space zero fibre (September 24, 2026).

Sources consulted on 12 October 2026 at repository revision fd4aeeb. Download the manuscript citations.