Tame Automorphism

A tame automorphism of a polynomial ring over a field is a ring automorphism fixing that is a finite function composition of coordinate changes of the form

where is a nonzero element of , is a polynomial independent of , and all other variables are fixed (Shestakov and Umirbaev 2004, p. 197). These changes are invertible because can be recovered by subtracting and dividing by . The tame automorphisms form a subgroup of the automorphism group of the polynomial ring. An invertible polynomial map is also called tame when the corresponding substitution ring automorphism is tame.

Every invertible polynomial map in two variables over a field is tame (Jung 1942, van der Kulk 1953). Jung's result for characteristic zero was extended by van der Kulk to arbitrary characteristic (Nguyen 2003, p. 209). In three variables, the Nagata automorphism is a wild automorphism over fields of characteristic zero.