The Shestakov–Umirbaev Theory and Nagata’s Conjecture

As discussed in the introduction of the book by van den Essen (Polynomial Automorphisms and the Jacobian Conjecture, Prog. Math., vol. 190, Birkhauser Verlag, Basel, 2000), Nagata conjectured that there exist wild polynomial automorphisms in three variables (Conjecture 1.1.1). In 2004, after van den Essen’s book was published, Shestakov–Umirbaev showed that the conjecture is true if the coefficient field is of characteristic zero. The purpose of this chapter is to give an introduction to the Shestakov–Umirbaev theory. We give a self-contained proof of a wildness criterion of polynomial automorphisms in three variables (Sect. 1.1.4). Using this criterion, we can easily check the wildness of Nagata’s famous automorphism (Exercise 11).