keller-birational

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 $k(x)=k(F)$ 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.

1. ↑ O.-H. Keller, Ganze Cremona-Transformationen, Monatsh. Math. Phys. 47 (1939) 299–306; zbMATH review by Kneser.