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In 2003, Shestakov-Umirbaev solved Nagata's conjecture on an automorphism ofa polynomial ring. In the present paper, we reconstruct their theory by usingthe "generalized Shestakov-Umirbaev inequality", which was recently given bythe author. As a consequence, we obtain a more precise tameness criterion forpolynomial automorphisms. In particular, we show that no tame automorphism of apolynomial ring admits a reduction of type IV.

Shestakov-Umirbaev reductions and Nagata's conjecture on a polynomial automorphism