# A stable coordinate in five variables

*Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).*

A polynomial \(f\in k[x_1,\ldots,x_n]\) is a **coordinate** if \(k[x_1,\ldots,x_n]=k[f,f_2,\ldots,f_n]\) for some polynomials \(f_2,\ldots,f_n\), and a **one-stable coordinate** if it is a coordinate of \(k[x_1,\ldots,x_n,w]\) for one further variable \(w\). The stable coordinate conjecture asked whether every one-stable coordinate is a coordinate. This lesson shows that the polynomial \(H\) of [A fourfold whose cylinder is affine space](a-fourfold-whose-cylinder-is-affine-space.md) is a one-stable coordinate of \(P=k[p,s,u,F,J]\) which is not a coordinate of \(P\). The complementary coordinates are written down explicitly. They come from a short transfer lemma: a polynomial of the form \(\lambda+d\,(S(\lambda)+ew)\) with \(d\) divisible by \(e\) is a coordinate, with an explicit partner obtained from a difference quotient.

We use Sections 2–4 of [A fourfold whose cylinder is affine space](a-fourfold-whose-cylinder-is-affine-space.md) (the polynomial \(H\), the exponential \(\Phi\) with \(\Phi(H)=H+p^3w\), and the coordinates \(L,M\)), and Theorem 9.1 of [Cancellation fails in dimension four](cancellation-fails-in-dimension-four.md#9-conclusion), which says that \(A=P/(H)\) is not a polynomial ring. The field \(k\) has characteristic zero.

Basic references are [OpenAI-cancellation] and [Dutta–Lahiri].

## 1. A transfer lemma

For a ring \(B\) and an indeterminate \(\lambda\), every \(S\in B[\lambda]\) has a **divided difference** \(\delta S\in B[\lambda,\mu]\), the polynomial with
\[
S(\lambda)-S(\mu)=(\lambda-\mu)\,\delta S(\lambda,\mu).
\]
It exists because \(\lambda^j-\mu^j=(\lambda-\mu)\sum_{i<j}\lambda^i\mu^{j-1-i}\).

**Lemma 1.1.** Let \(B\) be a noetherian ring, \(e,d'\in B\), \(d=ed'\), and \(S\in B[\lambda]\). In the polynomial ring \(B[\lambda,w]\) put
\[
\begin{gathered}
F=\lambda+d\,\bigl(S(\lambda)+ew\bigr),\\
G=w-d'\,\bigl(S(\lambda)+ew\bigr)\,\delta S(\lambda,F).
\end{gathered}
\]
Then \(B[\lambda,w]=B[F,G]\), and \(F,G\) are algebraically independent over \(B\). In particular \(F\) is a coordinate of \(B[\lambda,w]\) over \(B\).

**Proof.** Put \(\Phi=S(\lambda)+ew\), so that \(F=\lambda+d\Phi\). Then
\[
\begin{aligned}
&S(\lambda)-S(F)=(\lambda-F)\,\delta S(\lambda,F)\\
&\quad =-d\Phi\,\delta S(\lambda,F)=-e\,d'\Phi\,\delta S(\lambda,F),
\end{aligned}
\]
and therefore
\[
\begin{aligned}
eG&=ew-ed'\Phi\,\delta S(\lambda,F)\\
&=ew+S(\lambda)-S(F)=\Phi-S(F),
\end{aligned}
\]
that is, \(\Phi=S(F)+eG\).
Hence \(\Phi\in B[F,G]\), then \(\lambda=F-d\Phi\in B[F,G]\), and finally
\[
w=G+d'\Phi\,\delta S(\lambda,F)\in B[F,G].
\]
So the \(B\)-algebra map \(\theta\colon B[T_1,T_2]\to B[\lambda,w]\), \(T_1\mapsto F\), \(T_2\mapsto G\), is onto. Composing it with the isomorphism \(B[\lambda,w]\cong B[T_1,T_2]\), \(\lambda\mapsto T_1\), \(w\mapsto T_2\), gives a surjective ring endomorphism \(\sigma\) of the noetherian ring \(B[T_1,T_2]\). Such a map is injective: the kernels of \(\sigma^n\) increase and become constant, say from \(n\) on; if \(\sigma(x)=0\), write \(x=\sigma^n(y)\), so \(\sigma^{n+1}(y)=0\), hence \(\sigma^n(y)=0\), i.e. \(x=0\). So \(\theta\) is an isomorphism. \(\square\)

For example, with \(B=k[t]\), \(e=t\), \(d'=t\) and \(S=\lambda^2\), the polynomial \(F=\lambda+t^2\lambda^2+t^3w\) is a coordinate of \(k[t,\lambda,w]\) with partner \(G=w-t(\lambda^2+tw)(\lambda+F)\), although no automorphism of \(k[t][\lambda]\) alone moves \(\lambda\) to a polynomial congruent to \(\lambda+t^2\lambda^2\) modulo \(t^3\) (Exercise 4.2).

## 2. The shifted polynomial is a coordinate over B

Recall the notation of the cylinder lesson: \(x_0=s^2+u^3\),
\[
\begin{gathered}
L=x_0^2F-(1+2sx_0)J,\\
M=(1-2sx_0)F+4s^2J,
\end{gathered}
\]
a linear change of \(F,J\) over \(k[p,s,u]\) of determinant \(1\); so \(P=B[L]\) with \(B=k[p,s,u,M]\), a polynomial ring in four variables, and \(P[w]=B[L,w]\). In these coordinates
\[
\begin{gathered}
H=L-pu+p^2Q+p^4F^3,\\
Q=2x_0F^2-2sFJ-J^2,
\end{gathered}
\]
where \(F\) and \(J\) are linear in \(L\) over \(B\). Write \(Q=C+L\,Q_1\) with \(C=Q|_{L=0}\in B\) and \(Q_1\in B[L]\), and put \(L_*=pu-p^2C\in B\).

**Proposition 2.1.** The polynomial \(g=H+p^3w\) is a coordinate of \(B[L,w]\) over \(B\): there is \(G_2\in B[L,w]\) with \(B[L,w]=B[g,G_2]\).

**Proof.** We apply four \(B\)-algebra automorphisms of \(B[L,w]\), each of which moves one variable by an element of \(B\) or of the subring generated by the other variable over \(B\).

1. \(w\mapsto w-pF^3\). Since \(F\in B[L]\), this is an automorphism, and it sends \(g=L-pu+p^2Q+p^4F^3+p^3w\) to \(g_1=L-pu+p^2Q+p^3w\).
2. \(L\mapsto L_*+\lambda\), a translation by an element of \(B\); here \(\lambda\) is the new variable. Since \(L_*-pu+p^2C=0\),
\[
\begin{aligned}
g_1&=\lambda+p^2(L_*+\lambda)\,Q_1(L_*+\lambda)\\
&\quad +p^3w .
\end{aligned}
\]
3. \(L_*=p(u-pC)\), so \(p^2L_*Q_1(L_*+\lambda)\) equals \(p^3(u-pC)Q_1(L_*+\lambda)\). The automorphism \(w\mapsto w-(u-pC)Q_1(L_*+\lambda)\) turns \(g_1\) into
\[
\begin{gathered}
g_2=\lambda+p^2\bigl(S(\lambda)+pw\bigr),\\
S(\lambda)=\lambda\,Q_1(L_*+\lambda)\in B[\lambda].
\end{gathered}
\]
4. Lemma 1.1 with \(e=p\), \(d'=p\), \(d=p^2\) shows that \(g_2\) is a coordinate of \(B[\lambda,w]\) over \(B\).

Undoing the automorphisms of steps 1–3 carries the partner of \(g_2\) to a partner \(G_2\) of \(g\). \(\square\)

## 3. The theorem

**Theorem 3.1.** Let \(k\) be a field of characteristic zero. The polynomial \(H\in P=k[p,s,u,F,J]\) is a coordinate of \(P[w]=k^{[6]}\), but not a coordinate of \(P\). So the stable coordinate conjecture fails for polynomials in five variables, after one stabilization.

**Proof.** By Section 3 of the cylinder lesson, \(\Phi=\exp(w\Delta)\) is a \(k[w]\)-automorphism of \(P[w]\) with \(\Phi(H)=H+p^3w=g\). By Proposition 2.1, \(P[w]=B[g,G_2]=k[p,s,u,M,g,G_2]\), so
\[
\begin{aligned}
P[w]=k\bigl[&\Phi^{-1}(p),\Phi^{-1}(s),\Phi^{-1}(u),\\
&\Phi^{-1}(M),H,\Phi^{-1}(G_2)\bigr],
\end{aligned}
\]
and \(H\) is a coordinate of \(P[w]\). If \(H\) were a coordinate of \(P\), say \(P=k[H,f_2,\ldots,f_5]\), then \(A=P/(H)\cong k[f_2,\ldots,f_5]\) would be a polynomial ring in four variables, contrary to Theorem 9.1 of [Cancellation fails in dimension four](cancellation-fails-in-dimension-four.md#9-conclusion). \(\square\)

The theorem gives more than the failure of cancellation. Adjoining \(w\) to the hypersurface \(H=0\) gives affine five-space, as in the cylinder lesson; here the hypersurface \(H=0\) of \(\mathbb A^6=\operatorname{Spec}P[w]\) is even a coordinate hyperplane, while the hypersurface \(H=0\) of \(\mathbb A^5\) is not. The polynomial is a residual coordinate over \(k[p]\): after inverting \(p\) it is a coordinate over \(k[p,p^{-1}]\) (proof of Proposition 2.2 of the cylinder lesson), and modulo \(p\) it is congruent to the coordinate \(L\) (Section 4 there). Theorems of El Kahoui and Ouali and of Dutta and Lahiri [Dutta–Lahiri] show more generally that every residual coordinate over \(k[p]\) is a one-stable coordinate when \(k\) is algebraically closed; for the polynomial \(H\), Theorem 3.1 gives this over every field of characteristic zero.

## 4. Exercises

**Exercise 4.1.** In the example after Lemma 1.1, express \(\lambda\) and \(w\) as polynomials in \(F\) and \(G\).

**Exercise 4.2.** Let \(t\) be an indeterminate. Show that every automorphism of the \(k[t]\)-algebra \(k[t][\lambda]\) has the form \(\lambda\mapsto a\lambda+b\) with \(a\in k^\times\) and \(b\in k[t]\). Conclude that no such automorphism sends \(\lambda\) to a polynomial congruent to \(\lambda+t^2\lambda^2\) modulo \(t^3\).

**Exercise 4.3.** Show that a coordinate \(f\) of \(k[x_1,\ldots,x_n]\) has \(k[x_1,\ldots,x_n]/(f-c)\cong k^{[n-1]}\) for every \(c\in k\). Deduce that every fibre \(H=c\) of the polynomial of Theorem 3.1 becomes affine five-space after multiplication by a line.

**Exercise 4.4.** Show that in Lemma 1.1 the hypothesis that \(B\) is noetherian can be removed for the conclusion \(B[\lambda,w]=B[F,G]\), and that algebraic independence then follows by passing to a noetherian subring of \(B\) containing \(e\), \(d'\) and the coefficients of \(S\).

## 5. Solutions

**4.1.** With \(\Phi=\lambda^2+tw\), the proof of Lemma 1.1 gives \(\Phi=F^2+tG\), \(\lambda=F-t^2\Phi=F-t^2F^2-t^3G\), and \(w=G+t\Phi(\lambda+F)\), where \(\lambda+F=2F-t^2F^2-t^3G\). Substituting, \(w=G+t(F^2+tG)(2F-t^2F^2-t^3G)\).

**4.2.** An automorphism is determined by the image \(f(\lambda)\) of \(\lambda\), and its inverse by a polynomial \(g\) with \(g(f(\lambda))=\lambda\). Over the domain \(k[t]\), degrees multiply, so \(f\) has degree \(1\): \(f=a\lambda+b\). Then \(g(f)=\lambda\) forces \(a\) to be a unit of \(k[t]\), that is \(a\in k^\times\). Modulo \(t^3\), \(a\lambda+b\) has no term \(t^2\lambda^2\).

**4.3.** If \(k[x]=k[f,f_2,\ldots,f_n]\), then \(k[x]=k[f-c,f_2,\ldots,f_n]\), and the quotient by \(f-c\) is \(k[\bar f_2,\ldots,\bar f_n]\), a polynomial ring because \(f-c,f_2,\ldots,f_n\) are algebraically independent. Applied to \(H\) in \(P[w]\): \(P[w]/(H-c)\cong k^{[5]}\), and \(P[w]/(H-c)=(P/(H-c))[w]\).

**4.4.** The identities in the proof of Lemma 1.1 show surjectivity over any ring. If \(B_0\subseteq B\) is the subring generated by \(e\), \(d'\) and the coefficients of \(S\), it is noetherian, being a quotient of a polynomial ring over \(\mathbb Z\) in finitely many variables. Lemma 1.1 over \(B_0\) gives \(B_0[\lambda,w]=B_0[F,G]\) with \(F,G\) algebraically independent over \(B_0\), i.e. a \(B_0\)-algebra automorphism of \(B_0[\lambda,w]\) sending \(\lambda,w\) to \(F,G\). Its base change to \(B\) is an automorphism of \(B[\lambda,w]\) sending \(\lambda,w\) to \(F,G\).

## References

- [OpenAI-cancellation] OpenAI, An explicit failure of complex affine space cancellation, preprint, 23 September 2026. https://github.com/openai/math/tree/main/preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026
- [Dutta–Lahiri] A. K. Dutta, A. Lahiri, On residual and stable coordinates, Journal of Pure and Applied Algebra 225 (2021), 106707. https://arxiv.org/abs/1908.03549