# The cubic reduction of Bass–Connell–Wright and Yagzhev

- kind: reduction
- status: proved (evidence tier V3)
- year: 1982
- authors: A. V. Yagzhev (1980); H. Bass, E. H. Connell and D. Wright (1982), independently
- in one line: JC in every dimension ⇔ JC for x + H, H cubic homogeneous, in every dimension

**Statement.** Let $\operatorname{char}k=0$. (i) For $F=X+H$ with $H$ homogeneous of degree $d\ge2$, $\det JF\in k^{\times}$ if and only if $JH$ is nilpotent. (ii) Any Keller map of $k^n$ can be exchanged, by adding variables and composing with tame automorphisms, for a Keller map $X+H$ of some $k^N$ with $H$ cubic homogeneous, and the first is invertible if and only if the second is.

This is the standard normal form of the conjecture. At the price of more variables, any counterexample in any dimension can be put in the shape “identity plus a cubic homogeneous map with nilpotent Jacobian”. Yagzhev proved it in the Siberian Mathematical Journal in 1980. Bass, Connell and Wright proved it independently, in the 1982 Bulletin survey that also contains Gabber's degree bound and the [formal-inverse expansion](/wiki/Statement:formal-inverse.md).[1] Maps of this shape are often called Yagzhev maps.

The reduction is stable: it holds only over all dimensions at once, because the dimension grows. So [the reduction constrains no single JCₙ, and it says nothing about the plane](/wiki/Statement:jc-n.md). It is the theorem behind the equivalence recorded on [the cubic-homogeneous conjecture](/wiki/Statement:cubic-homogeneous-jc.md). With [Wang's theorem that quadratic Keller maps are invertible](/wiki/Statement:wang-degree-2.md), it makes degree 3 exactly the threshold.

After July 2026 the theorem still holds, but its target is false. Thompson ran the construction on Alpöge's map and obtained an explicit 24-variable example: see [cubic-homogeneous counterexamples](/wiki/Statement:cubic-homogeneous-counterexamples.md). The sharper normal forms of [Drużkowski](/wiki/Statement:druzkowski-cubic-linear.md) and [de Bondt–van den Essen](/wiki/Statement:de-bondt-van-den-essen-symmetric.md) start from this one.

## Relations

- related to [jc-n](/wiki/Statement:jc-n.md) — So the reduction constrains no single JCₙ, and it says nothing about the plane.
- related to [wang-degree-2](/wiki/Statement:wang-degree-2.md) — With Wang's theorem that quadratic Keller maps are invertible, it makes degree 3 exactly the threshold.

## 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. A. V. Yagzhev, Keller's problem, Siberian Math. J. 21 (1980) 747–754, [doi:10.1007/BF00973892](https://doi.org/10.1007/BF00973892).

## Relations to this page

- used by [cubic-homogeneous-counterexamples](/wiki/Statement:cubic-homogeneous-counterexamples.md) — On 21 July 2026 Thompson ran the Bass–Connell–Wright reduction on Alpöge's map by hand.
- used by [de-bondt-van-den-essen-symmetric](/wiki/Statement:de-bondt-van-den-essen-symmetric.md) — The symmetric reduction turns the Jacobian conjecture into a question about gradient maps of Hessian-nilpotent quartics. The construction takes a cubic-homogeneous Keller map in n variables and produces a Hessian-nilpotent quartic in $2n$ variables.
- used by [druzkowski-cubic-linear](/wiki/Statement:druzkowski-cubic-linear.md) — 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. The proof starts from the Bass–Connell–Wright and Yagzhev cubic-homogeneous form.

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