Difference between revisions
wang-degree-2
Status and infobox
- status:
—→ proved - tier:
—→ V3 - kind:
—→ theorem - year:
—→ 1980 - 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. - short:
—→ Every Keller map of degree at most 2 is an automorphism, in every dimension
Relations
- + special case of jc-n: 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.
Source +17 −0
+---
+title: Wang's theorem: quadratic Keller maps are invertible
+short: Every Keller map of degree at most 2 is an automorphism, in every dimension
+kind: theorem
+status: proved
+tier: V3
+year: 1980
+authors: Stuart Sui-Sheng Wang
+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.<ref>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).</ref> It says [[special-case-of::Statement:jc-n|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 [[Statement:bcw-cubic-homogeneous|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.
+
+<references />
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Revision 283 as rendered
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, . 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 the map has derivative 1, yet sends 0 and 1 to 0.
- ↑ S. S.-S. Wang, A Jacobian criterion for separability, J. Algebra 65 (1980) 453–494, doi:10.1016/0021-8693(80)90233-1.