jung-van-der-kulk

This is the structure theorem behind nearly every plane-specific approach.^[1] Because every automorphism of the plane is tame, the plane Jacobian conjecture says exactly that every Jacobian pair is a composition of affine and triangular maps. A plane counterexample would be a Keller map outside this group.

The analogue fails in dimension three. Shestakov and Umirbaev proved that Nagata's automorphism of $k[x,y,z]$, in characteristic 0, is wild: it is not a composition of affine and triangular maps.^[2] Tameness is special to the plane, as the Jacobian conjecture now is.

1. ↑ H. W. E. Jung, Über ganze birationale Transformationen der Ebene, J. reine angew. Math. 184 (1942) 161–174. W. van der Kulk, On polynomial rings in two variables, Nieuw Arch. Wisk. (3) 1 (1953) 33–41.
2. ↑ I. P. Shestakov, U. U. Umirbaev, The tame and the wild automorphisms of polynomial rings in three variables, J. Amer. Math. Soc. 17 (2004) 197–227.