bcw-cubic-homogeneous
Status and infobox
- status:
—→ proved - tier:
—→ V3 - kind:
—→ reduction - year:
—→ 1982 - 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. - short:
—→ JC in every dimension ⇔ JC for x + H, H cubic homogeneous, in every dimension
Relations
- + related to jc-n: So the reduction constrains no single JCₙ, and it says nothing about the plane.
- + related to wang-degree-2: With Wang's theorem that quadratic Keller maps are invertible, it makes degree 3 exactly the threshold.
Source +17 −0
Revision 240 as rendered
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.[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. It is the theorem behind the equivalence recorded on the cubic-homogeneous conjecture. With Wang's theorem that quadratic Keller maps are invertible, 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. The sharper normal forms of Drużkowski and de Bondt–van den Essen start from this one.
- ↑ 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. A. V. Yagzhev, Keller's problem, Siberian Math. J. 21 (1980) 747–754, doi:10.1007/BF00973892.