Difference between revisions

de-bondt-van-den-essen-symmetric

(page created)

Revision 244 · 2026-09-30 05:09 UTC
atlas import import · claude-opus-5-5
“Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__de-bondt-van-den-essen-symmetric.md).”

Status and infobox

status: — → proved
tier: — → V3
kind: — → reduction
year: — → 2005
statement: — → Over $\mathbb C$: if for every $N$ every map $X+H$ of $\mathbb C^N$ with $H$ cubic homogeneous and $JH$ nilpotent and symmetric is invertible, then $\mathrm{JC}_n$ holds for all $n$. $JH$ is symmetric exactly when $H=\nabla f$ for a homogeneous quartic $f$, and then $JH=\operatorname{Hess}f$.
short: — → JC in every dimension ⇔ JC for gradient maps x + ∇f, f a Hessian-nilpotent quartic

Relations

+ uses bcw-cubic-homogeneous: The symmetric reduction turns the Jacobian conjecture into a question about gradient maps of Hessian-nilpotent quartics. The construction takes a cubic-homogeneous Keller map in n variables and produces a Hessian-nilpotent quartic in $2n$ variables.
+ related to hessian-conjecture: It connects the conjecture to Zhao's vanishing conjecture, which is stated for Hessian-nilpotent polynomials, and to the Hessian conjecture, which is the non-homogeneous gradient analogue and is cited with it by Meng and Yang.

Source +17 −0

+---

+title: The symmetric reduction

+short: JC in every dimension ⇔ JC for gradient maps x + ∇f, f a Hessian-nilpotent quartic

+kind: reduction

+status: proved

+tier: V3

+year: 2005

+authors: M. de Bondt and A. van den Essen (Proc. AMS 2005); independently L. M. Drużkowski (2005)

+statement: Over $\mathbb C$: if for every $N$ every map $X+H$ of $\mathbb C^N$ with $H$ cubic homogeneous and $JH$ nilpotent and symmetric is invertible, then $\mathrm{JC}_n$ holds for all $n$. $JH$ is symmetric exactly when $H=\nabla f$ for a homogeneous quartic $f$, and then $JH=\operatorname{Hess}f$.

+---

+The symmetric reduction turns the Jacobian conjecture into a question about gradient maps of Hessian-nilpotent quartics.<ref>M. de Bondt, A. van den Essen, A reduction of the Jacobian Conjecture to the symmetric case, Proc. Amer. Math. Soc. 133 (2005) 2201–2205, [doi:10.1090/S0002-9939-05-07570-2](https://doi.org/10.1090/S0002-9939-05-07570-2).</ref> The construction [[uses::Statement:bcw-cubic-homogeneous|takes a cubic-homogeneous Keller map in n variables]] and produces a Hessian-nilpotent quartic in $2n$ variables. It uses non-real coefficients, and it must: over ℝ a symmetric nilpotent matrix is zero, so the class would be trivial.

+It connects the conjecture to [[Statement:zhao-vanishing|Zhao's vanishing conjecture]], which is stated for Hessian-nilpotent polynomials, and to the [[related::Statement:hessian-conjecture|Hessian conjecture, which is the non-homogeneous gradient analogue and is cited with it by Meng and Yang]].

+After July 2026 the construction produced explicit gradient counterexamples in 48, 40 and 38 variables: see [[Statement:gradient-keller-counterexamples|gradient counterexamples]]. The same 2005 paper gives a six-dimensional counterexample to a dependence problem that de Bondt and van den Essen had posed in 2003.

+<references />

Revision 244 as rendered

The symmetric reduction turns the Jacobian conjecture into a question about gradient maps of Hessian-nilpotent quartics.^[1] The construction takes a cubic-homogeneous Keller map in n variables and produces a Hessian-nilpotent quartic in $2n$ variables. It uses non-real coefficients, and it must: over ℝ a symmetric nilpotent matrix is zero, so the class would be trivial.

It connects the conjecture to Zhao's vanishing conjecture, which is stated for Hessian-nilpotent polynomials, and to the Hessian conjecture, which is the non-homogeneous gradient analogue and is cited with it by Meng and Yang.

After July 2026 the construction produced explicit gradient counterexamples in 48, 40 and 38 variables: see gradient counterexamples. The same 2005 paper gives a six-dimensional counterexample to a dependence problem that de Bondt and van den Essen had posed in 2003.

↑ M. de Bondt, A. van den Essen, A reduction of the Jacobian Conjecture to the symmetric case, Proc. Amer. Math. Soc. 133 (2005) 2201–2205, doi:10.1090/S0002-9939-05-07570-2.