# The stable equivalence of the Jacobian and Dixmier conjectures

- kind: theorem
- status: proved (evidence 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)
- in one line: JC₂ₙ ⇒ DCₙ, so JC in every dimension ⇔ DC in every rank

**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.[1] 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.[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](/wiki/Statement:poisson-conjecture.md).

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](/wiki/Statement:dixmier-conjecture.md).

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.

## Relations

- related to [poisson-conjecture](/wiki/Statement:poisson-conjecture.md) — Adjamagbo and van den Essen gave a third proof, purely algebraic, through the Poisson conjecture, which isolates the symplectic step as its own statement.
- related to [dixmier-conjecture](/wiki/Statement:dixmier-conjecture.md) — 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.

## References

1. 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.
2. Y. Tsuchimoto, Endomorphisms of Weyl algebra and p-curvatures, Osaka J. Math. 42 (2005) 435–452.

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