# Jung–van der Kulk: automorphisms of the plane are tame

- kind: theorem
- status: proved (evidence tier V3)
- year: 1942
- authors: H. W. E. Jung (characteristic 0, 1942); W. van der Kulk (any field, 1953)
- in one line: Every automorphism of the plane is a composition of affine and triangular maps

**Statement.** For any field $k$, the group of polynomial automorphisms of $\mathbb A^2_k$ is generated by the affine maps and the triangular maps $(ax+P(y),\,by+c)$, and it is their amalgamated free product over their intersection.

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](/wiki/Statement:jc-plane.md). 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.

## Relations

- related to [jc-plane](/wiki/Statement:jc-plane.md) — This is the structure theorem behind nearly every plane-specific approach. 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. [1]

## References

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.

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