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

Abstract

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

Weighted multidegrees of polynomial automorphisms over a domain

“…For F, G ∈ Aut k k[x], we say that the pair (F, G) satisfies the Shestakov-Umirbaev condition for the weight w if the following conditions hold (cf. [13] Here, we recall that f 1 ≈ f 2 denotes that f 1 and f 2 are linearly dependent over k for each f 1 , f 2 ∈ k[x]. For each F ∈ Aut k k[x] and σ ∈ S 3 , we define F σ = (f σ(1) , f σ(2) , f σ(3) ).…”

“…[13,Theorem 4.2]). Here, we regard Γ as a subgroup of Q ⊗ Z Γ which has a structure of totally ordered additive group induced from Γ: (P1) (g Now, let us prove Lemma 9.2 by contradiction.…”

“…To prove Theorem 1.9, we use the generalized Shestakov-Umirbaev theory [12], [13]. For the convenience of the reader, we give a short introduction to this theory.…”

“…By definition, we have deg w f 3 = w 3 and f w 3 = αx 3 + p ′ for such F , where p ′ := p w if deg w p = w 3 , and p ′ := 0 otherwise. Recently, the author [12], [13] generalized the Shestakov-Umirbaev theory. By means of this theory, we prove the following theorem in Section 9.…”

“…The rest of this section is devoted to the proof of Theorem 1.10. To prove (ii) of this theorem, we need the following version of the Shestakov-Umirbaev inequality (see [13,Section 3] for detail). Let S = {f, g} be a subset of k [x] such that f and g are algebraically independent over k, and p a nonzero element of k [S].…”