The Jacobian conjecture for cubic-homogeneous maps

For a map $X+H$ with $H$ homogeneous of degree at least 2, the Keller condition is equivalent to $JH$ being nilpotent, and then $\det JF\equiv1$. By the reduction of Bass, Connell and Wright, and independently Yagzhev, any Keller map becomes a cubic-homogeneous one after adding variables and composing with tame automorphisms, and the original map is invertible exactly when the new one is. So the cubic-homogeneous case over all dimensions is equivalent to the Jacobian conjecture over all dimensions.^[1] The equivalence raises the dimension, so it says nothing about any single $N$.

It is false. The equivalence and the July 2026 counterexample make it false in some dimension. Explicit witnesses followed within weeks: see cubic-homogeneous counterexamples in 24, 20 and 19 variables. The degree 3 is sharp from below. Wang's theorem makes every quadratic Keller map invertible in every dimension, so no reduction to degree 2 can exist.

Small dimensions are known: Hubbers proved the case $N=4$ in 1994, and padding gives every smaller $N$. The smallest dimension with a cubic-homogeneous counterexample therefore lies between 5 and 19, and where it lies is open.

Two sharper normal forms exist: cubic-linear maps, in which each component of $H$ is the cube of a linear form, and symmetric maps, in which $H$ is a gradient. By Drużkowski (2005) a counterexample can take either form, but never both at once.

1. ↑ H. Bass, E. H. Connell, D. Wright, The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. 7 (1982) 287–330.