# Drużkowski's reduction to cubic-linear maps

- kind: reduction
- status: proved (evidence tier V3)
- year: 1983
- authors: L. M. Drużkowski
- in one line: JC in every dimension ⇔ JC for x + (Ax)^{*3}, in every dimension

**Statement.** Over $\mathbb C$: if for every $N$ every map $F=X+(AX)^{*3}$ of $\mathbb C^N$ with $\det JF\equiv1$ is invertible, then $\mathrm{JC}_n$ holds for all $n$. Here $A\in M_N(\mathbb C)$ and $(AX)^{*3}$ is the vector of cubes of the coordinates of $AX$.

Drużkowski refined the cubic normal form further: after adding variables, each nonlinear component can be taken to be the cube of a linear form. The Keller condition then says that the matrix $3\,\mathrm{diag}\big((AX)^{*2}\big)A$ is nilpotent, and the problem becomes one about matrices.[1] The proof [starts from the Bass–Connell–Wright and Yagzhev cubic-homogeneous form](/wiki/Statement:bcw-cubic-homogeneous.md). In 2001 Drużkowski showed one may also assume $A^2=0$. Like its parent, the reduction raises the dimension, so [it bears on the all-dimensions conjecture only, not on any fixed dimension](/wiki/Statement:jc-n.md).

In 2005 Drużkowski proved that cubic-linear maps with symmetric Jacobian and constant nonzero determinant are tame automorphisms. 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](/wiki/Statement:de-bondt-van-den-essen-symmetric.md).[2]

Jointly with the 2026 counterexample, the reduction guarantees a non-invertible cubic-linear Keller map in some dimension. None has been exhibited explicitly in the sources the research found.

## Relations

- uses [bcw-cubic-homogeneous](/wiki/Statement:bcw-cubic-homogeneous.md) — The Keller condition then says that the matrix $3\,\mathrm{diag}\big((AX)^{*2}\big)A$ is nilpotent, and the problem becomes one about matrices. The proof starts from the Bass–Connell–Wright and Yagzhev cubic-homogeneous form. [1]
- related to [jc-n](/wiki/Statement:jc-n.md) — Like its parent, the reduction raises the dimension, so it bears on the all-dimensions conjecture only, not on any fixed dimension.
- related to [de-bondt-van-den-essen-symmetric](/wiki/Statement:de-bondt-van-den-essen-symmetric.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. [2]

## References

1. L. M. Drużkowski, An effective approach to Keller's Jacobian Conjecture, Math. Ann. 264 (1983) 303–313, [doi:10.1007/BF01459126](https://doi.org/10.1007/BF01459126).
2. L. M. Drużkowski, [The Jacobian Conjecture: symmetric reduction and solution in the symmetric cubic linear case](https://eudml.org/doc/280721), Ann. Polon. Math. 87 (2005) 83–92.

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