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Wild Automorphism

A wild automorphism of a polynomial ring over a field K is a ring automorphism fixing K that is not a tame automorphism. Thus it cannot be obtained by a finite function composition of invertible changes of a single coordinate that are linear in that coordinate (Shestakov and Umirbaev 2004, p. 197).

The Nagata automorphism is wild over every field of characteristic zero (Shestakov and Umirbaev 2003, p. 12563; 2004, p. 224). In contrast, all invertible polynomial maps in two variables over a field are tame automorphisms.

See also

Invertible Polynomial Map, Nagata Automorphism, Polynomial Ring, Tame Automorphism

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References

Shestakov, I. P. and Umirbaev, U. U. "The Nagata Automorphism Is Wild." Proc. Natl. Acad. Sci. USA 100, 12561-12563, 2003. https://doi.org/10.1073/pnas.1735483100.Shestakov, I. P. and Umirbaev, U. U. "The Tame and the Wild Automorphisms of Polynomial Rings in Three Variables." J. Amer. Math. Soc. 17, 197-227, 2004. https://doi.org/10.1090/S0894-0347-03-00440-5.

Cite this as:

Weisstein, Eric W. "Wild Automorphism." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WildAutomorphism.html

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