Keller's birational theorem
A birational Keller map is an automorphism
Keller proved this in the paper where he posed the conjecture. He worked over the integers with functional determinant 1, and showed that if the map is rationally invertible, its inverse is polynomial with integer coefficients. The proof used homology.[1] In modern terms, the conjecture holds for Keller maps of field degree 1, so Keller's theorem is the Jacobian conjecture restricted to birational maps.
The trivial extension is Galois, so Keller's theorem is the degree-1 case of Campbell's Galois criterion.
It is the first wall around every counterexample. A non-invertible Keller map has field degree at least 2, and in fact at least 3 (see the degree spectrum). Alpöge's map has degree 3.
- ↑ O.-H. Keller, Ganze Cremona-Transformationen, Monatsh. Math. Phys. 47 (1939) 299–306; zbMATH review by Kneser.
Relations
- special case of The Jacobian conjecture in dimension n refuted · V0
The proof used homology. In modern terms, the conjecture holds for Keller maps of field degree 1, so Keller's theorem is the Jacobian conjecture restricted to birational maps. [1]
- special case of Campbell's theorem: Galois Keller maps are automorphisms proved · V3
The trivial extension is Galois, so Keller's theorem is the degree-1 case of Campbell's Galois criterion.
Relations to this page
- used by The degree spectrum of ℂ³ proved · V4
Degrees 1 and 2 force an automorphism. Degree 1 is Keller's birational theorem.
- used by No plane Keller map has prime field degree proved · V4
It then applies Keller's birational theorem, which closes degree 1.
- used by For a Keller map, invertible, injective and proper are the same proved · V3
(iv) ⇒ (i). This is Keller's birational theorem, which turns a birational Keller map into an automorphism.
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