Nagata Automorphism

The Nagata automorphism is the invertible polynomial map in three variables introduced by Nagata (1972, pp. 41-42) and defined by

where and the coefficients lie in a field of characteristic zero. The polynomial satisfies , so the inverse function is

Shestakov and Umirbaev (2003, p. 12563; 2004, p. 224) proved that this map is a wild automorphism. It cannot be expressed as a finite function composition of the coordinate changes generating the tame automorphisms. This distinguishes three variables from two, in which every invertible polynomial map is a tame automorphism.

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Nagata, M. On Automorphism Group of

. Tokyo, Japan: Kinokuniya, pp. 41-42, 1972. https://hdl.handle.net/2433/84910.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.Trott, M. "Solution of Equations." §1.5 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28815-5. https://www.mathematicaguidebooks.org/.