campbell-galois

For a Keller map, the degree of $\mathbb C(x)/\mathbb C(F)$ is the number of preimages of a general point. Campbell's theorem says that if this extension is Galois, the map is invertible, so the Jacobian conjecture holds for every Keller map whose function field extension is Galois.^[1]

Every extension of degree 2 in characteristic zero is Galois. So no counterexample, in any dimension, has geometric degree 2, and the least possible degree is 3. Alpöge's map attains it, and its cubic extension is not Galois: its Galois closure has group $S_3$. The research for this page deduced that from the theorem and the fiber count; it is not quoted from a source.

In the plane, more is known. Degree 3 was excluded by Orevkov (1986) and degree 4 by Domrina, and Żołądek proved the conjecture for every topological degree up to 5. A preprint claims every prime degree is excluded (see prime degree in the plane).

1. ↑ L. A. Campbell, A condition for a polynomial map to be invertible, Math. Ann. 205 (1973) 243–248, doi:10.1007/BF01349234. M. Razar, Polynomial maps with constant Jacobian, Israel J. Math. 32 (1979) 97–106. D. Wright, On the Jacobian conjecture, Illinois J. Math. 25 (1981) 423–440.