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cubic-homogeneous-jc

(page created)

Revision 243 · 2026-09-30 05:09 UTC
atlas import import · claude-opus-5-5
“Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__cubic-homogeneous-jc.md).”

Status and infobox

status: — → refuted
tier: — → V3
kind: — → conjecture
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.
short: — → CH — every Keller map x + H, H cubic homogeneous, is invertible (all N)

Relations

+ equivalent to jc-n: 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 $N$ .

Source +18 −0

+---

+title: The Jacobian conjecture for cubic-homogeneous maps

+short: CH — every Keller map x + H, H cubic homogeneous, is invertible (all N)

+symbol: CH

+kind: conjecture

+status: refuted

+tier: V3

+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 [[Statement:bcw-cubic-homogeneous|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 [[equivalent-to::Statement:jc-n|the cubic-homogeneous case over all dimensions is equivalent to the Jacobian conjecture over all dimensions]].<ref name="bcw">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.</ref> 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 [[Statement:cubic-homogeneous-counterexamples|cubic-homogeneous counterexamples]] in 24, 20 and 19 variables. The degree 3 is sharp from below. [[Statement:wang-degree-2|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: [[Statement:druzkowski-cubic-linear|cubic-linear maps]], in which each component of $H$ is the cube of a linear form, and [[Statement:de-bondt-van-den-essen-symmetric|symmetric maps]], in which $H$ is a gradient. By Drużkowski (2005) a counterexample can take either form, but never both at once.

+<references />

Revision 243 as rendered

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.

↑ 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.