Wang's theorem: quadratic Keller maps are invertible

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.

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, 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.

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