druzkowski-cubic-linear

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. 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.

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.^[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.

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