Difference between revisions
jung-van-der-kulk
Status and infobox
- status:
—→ proved - tier:
—→ V3 - kind:
—→ theorem - year:
—→ 1942 - 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. - short:
—→ Every automorphism of the plane is a composition of affine and triangular maps
Relations
- + related to jc-plane: 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.
Source +15 −0
+---
+title: Jung–van der Kulk: automorphisms of the plane are tame
+short: Every automorphism of the plane is a composition of affine and triangular maps
+kind: theorem
+status: proved
+tier: V3
+year: 1942
+authors: H. W. E. Jung (characteristic 0, 1942); W. van der Kulk (any field, 1953)
+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.<ref>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.</ref> Because every automorphism of the plane is tame, [[related::Statement:jc-plane|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.<ref>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.</ref> Tameness is special to the plane, as the Jacobian conjecture now is.
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+<references />
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Revision 266 as rendered
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 , 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.
- ↑ 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.
- ↑ 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.