# Wang's theorem: quadratic Keller maps are invertible

- kind: theorem
- status: proved (evidence tier V3)
- year: 1980
- authors: Stuart Sui-Sheng Wang
- in one line: Every Keller map of degree at most 2 is an automorphism, in every dimension

**Statement.** Let $\operatorname{char}k=0$ and $F\in k[x_1,\dots,x_n]^n$ with $\deg F\le2$ and $\det JF\in k^{\times}$. Then $F$ is a polynomial automorphism.

Wang's theorem appears in a paper on a Jacobian criterion for separability.[1] It says [the Jacobian conjecture holds, in every dimension, for maps of degree at most two](/wiki/Statement:jc-n.md).

Over ℂ there is a one-line reason. For a quadratic map, $F(a)-F(b)=JF\big(\tfrac{a+b}2\big)(a-b)$. An invertible Jacobian therefore forces injectivity, and an injective polynomial self-map of ℂⁿ is an automorphism. The research for this page checked the midpoint identity. It is not Wang's proof.

With the [cubic reduction](/wiki/Statement:bcw-cubic-homogeneous.md), Wang's theorem makes degree 3 the exact threshold: every counterexample has degree at least 3, and the reduction needs no more. The hypothesis on the characteristic matters. Over $\overline{\mathbb F}_2$ the map $x\mapsto x+x^2$ has derivative 1, yet sends 0 and 1 to 0.

## Relations

- special case of [jc-n](/wiki/Statement:jc-n.md) — Wang's theorem appears in a paper on a Jacobian criterion for separability. It says the Jacobian conjecture holds, in every dimension, for maps of degree at most two. [1]

## References

1. S. S.-S. Wang, A Jacobian criterion for separability, J. Algebra 65 (1980) 453–494, [doi:10.1016/0021-8693(80)90233-1](https://doi.org/10.1016/0021-8693(80)90233-1).

## Relations to this page

- related to [bcw-cubic-homogeneous](/wiki/Statement:bcw-cubic-homogeneous.md) — With Wang's theorem that quadratic Keller maps are invertible, it makes degree 3 exactly the threshold.

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