# The Jacobian conjecture for cubic-homogeneous maps

- kind: conjecture
- status: refuted (evidence tier V3)
- in one line: CH — every Keller map x + H, H cubic homogeneous, is invertible (all N)

**Statement.** For every $N$, every map $F=X+H:k^N\to k^N$ with $H$ homogeneous of degree 3 and $\det JF\in k^\times$ (equivalently, $JH$ nilpotent) has a polynomial inverse.

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](/wiki/Statement:bcw-cubic-homogeneous.md), 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](/wiki/Statement:jc-n.md).[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](/wiki/Statement:cubic-homogeneous-counterexamples.md) in 24, 20 and 19 variables. The degree 3 is sharp from below. [Wang's theorem](/wiki/Statement:wang-degree-2.md) 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](/wiki/Statement:druzkowski-cubic-linear.md), in which each component of $H$ is the cube of a linear form, and [symmetric maps](/wiki/Statement:de-bondt-van-den-essen-symmetric.md), in which $H$ is a gradient. By Drużkowski (2005) a counterexample can take either form, but never both at once.

## Relations

- equivalent to [jc-n](/wiki/Statement:jc-n.md) — So the cubic-homogeneous case over all dimensions is equivalent to the Jacobian conjecture over all dimensions. The equivalence raises the dimension, so it says nothing about any single $N$. [1]

## References

1. H. Bass, E. H. Connell, D. Wright, [The Jacobian conjecture: reduction of degree and formal expansion of the inverse](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-7/issue-2/The-Jacobian-conjecture--Reduction-of-degree-and-formal-expansion/bams/1183549636.full), Bull. Amer. Math. Soc. 7 (1982) 287–330.

## Relations to this page

- refuted by [cubic-homogeneous-counterexamples](/wiki/Statement:cubic-homogeneous-counterexamples.md) — These maps refute the cubic-homogeneous Jacobian conjecture explicitly, not only by the Bass–Connell–Wright equivalence.

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