# The symmetric reduction

- kind: reduction
- status: proved (evidence tier V3)
- year: 2005
- authors: M. de Bondt and A. van den Essen (Proc. AMS 2005); independently L. M. Drużkowski (2005)
- in one line: JC in every dimension ⇔ JC for gradient maps x + ∇f, f a Hessian-nilpotent quartic

**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.[1] The construction [takes a cubic-homogeneous Keller map in n variables](/wiki/Statement:bcw-cubic-homogeneous.md) 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](/wiki/Statement:zhao-vanishing.md), 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](/wiki/Statement:hessian-conjecture.md).

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

## Relations

- uses [bcw-cubic-homogeneous](/wiki/Statement:bcw-cubic-homogeneous.md) — 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. [1]
- related to [hessian-conjecture](/wiki/Statement:hessian-conjecture.md) — 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.

## References

1. 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).

## Relations to this page

- related to [druzkowski-cubic-linear](/wiki/Statement:druzkowski-cubic-linear.md) — So a counterexample can be put in cubic-linear form or in the symmetric form of de Bondt and van den Essen, but never in both at once.
- used by [gradient-keller-counterexamples](/wiki/Statement:gradient-keller-counterexamples.md) — The symmetric construction of de Bondt and van den Essen turns a cubic-homogeneous Keller map in $n$ variables into a gradient map of a Hessian-nilpotent quartic in $2n$ variables.
- used by [zhao-vanishing](/wiki/Statement:zhao-vanishing.md) — By the symmetric reduction, the Jacobian conjecture reduces to gradient maps $z-\nabla P$ with $P$ a Hessian-nilpotent homogeneous quartic (Zhao's argument starts from the de Bondt–van den Essen symmetric form).

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