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 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 (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, 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. \(\square\)

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