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tsuchimoto-bkk

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Revision 282 · 2026-09-30 05:09 UTC
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“Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__tsuchimoto-bkk.md).”

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+---
+title: The stable equivalence of the Jacobian and Dixmier conjectures
+short: JC₂ₙ ⇒ DCₙ, so JC in every dimension ⇔ DC in every rank
+kind: theorem
+status: proved
+tier: V3
+year: 2005
+authors: Y. Tsuchimoto (2005); A. Belov-Kanel and M. Kontsevich (2005, published 2007); V. V. Bavula (2005); K. Adjamagbo and A. van den Essen (2007)
+statement: For every $n\ge1$ and every field of characteristic $0$, $\mathrm{JC}_{2n}\Rightarrow\mathrm{DC}_n$. With $\mathrm{DC}_n\Rightarrow\mathrm{JC}_n$, this gives $(\forall n\ \mathrm{JC}_n)\iff(\forall n\ \mathrm{DC}_n)$.
+---
+The proof reduces modulo large primes. In characteristic $p$ the center of the Weyl algebra $A_n$ is a polynomial ring in $2n$ variables, the $p$-th powers of the generators, and it carries a Poisson bracket. An endomorphism of $A_n$ induces a symplectic map of that $2n$-dimensional space, whose Jacobian is 1. The Jacobian conjecture in dimension $2n$ makes that map invertible, and the invertibility lifts back.<ref name="bkk">A. Belov-Kanel, M. Kontsevich, [The Jacobian conjecture is stably equivalent to the Dixmier conjecture](https://arxiv.org/abs/math/0512171), Moscow Math. J. 7 (2007) 209–218, Theorem 1.</ref> Tsuchimoto phrased the argument with $p$-curvatures, and Belov-Kanel and Kontsevich call it antiquantization. Belov-Kanel and Kontsevich note that Tsuchimoto's earlier paper contains their main result.<ref>Y. Tsuchimoto, Endomorphisms of Weyl algebra and p-curvatures, Osaka J. Math. 42 (2005) 435–452.</ref> Bavula gave a short proof by inversion formulas. Adjamagbo and van den Essen gave a third proof, purely algebraic, through the [[related::Statement:poisson-conjecture|Poisson conjecture, which isolates the symplectic step as its own statement]].
 
+The theorem supplies the implication recorded on the [[related::Statement:dixmier-conjecture|Dixmier family as “DCₙ reduces to JC₂ₙ”, which with DCₙ ⇒ JCₙ sandwiches DCₙ between JCₙ and JC₂ₙ]].
+ 
+After July 2026 its hypothesis is false in every dimension $2n\ge4$, so it says nothing about DCₙ for $n\ge2$. Its only live instance is $n=1$: JC₂ ⇒ DC₁. The open plane conjecture now controls Dixmier's original problem.
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Revision 282 as rendered

The proof reduces modulo large primes. In characteristic pp the center of the Weyl algebra AnA_n is a polynomial ring in 2n2n variables, the pp-th powers of the generators, and it carries a Poisson bracket. An endomorphism of AnA_n induces a symplectic map of that 2n2n-dimensional space, whose Jacobian is 1. The Jacobian conjecture in dimension 2n2n makes that map invertible, and the invertibility lifts back.[1] Tsuchimoto phrased the argument with pp-curvatures, and Belov-Kanel and Kontsevich call it antiquantization. Belov-Kanel and Kontsevich note that Tsuchimoto's earlier paper contains their main result.[2] Bavula gave a short proof by inversion formulas. Adjamagbo and van den Essen gave a third proof, purely algebraic, through the Poisson conjecture, which isolates the symplectic step as its own statement.

The theorem supplies the implication recorded on the Dixmier family as “DCₙ reduces to JC₂ₙ”, which with DCₙ ⇒ JCₙ sandwiches DCₙ between JCₙ and JC₂ₙ.

After July 2026 its hypothesis is false in every dimension 2n≥42n\ge4, so it says nothing about DCₙ for n≥2n\ge2. Its only live instance is n=1n=1: JC₂ ⇒ DC₁. The open plane conjecture now controls Dixmier's original problem.

  1. ↑ A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier conjecture, Moscow Math. J. 7 (2007) 209–218, Theorem 1.
  2. ↑ Y. Tsuchimoto, Endomorphisms of Weyl algebra and p-curvatures, Osaka J. Math. 42 (2005) 435–452.