druzkowski-cubic-linear
Status and infobox
- status:
—→ proved - tier:
—→ V3 - kind:
—→ reduction - year:
—→ 1983 - 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 $AX$. - short:
—→ JC in every dimension ⇔ JC for x + (Ax)^{*3}, in every dimension
Relations
- + uses bcw-cubic-homogeneous: The Keller condition then says that the matrix is nilpotent, and the problem becomes one about matrices. The proof starts from the Bass–Connell–Wright and Yagzhev cubic-homogeneous form.
- + related to jc-n: 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: 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.
Source +17 −0
Revision 249 as rendered
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 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 . 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.
- ↑ L. M. Drużkowski, An effective approach to Keller's Jacobian Conjecture, Math. Ann. 264 (1983) 303–313, doi:10.1007/BF01459126.
- ↑ L. M. Drużkowski, The Jacobian Conjecture: symmetric reduction and solution in the symmetric cubic linear case, Ann. Polon. Math. 87 (2005) 83–92.