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Shestakov-Umirbaev reductions and Nagata’s conjecture on a polynomial automorphism. (English) Zbl 1210.14072

A long standing problem in the algebraic theory of polynomial automorphisms has been to decide whether the automorphism group of a polynomial ring in three variables over a field is generated by the so-called tame automorphisms. In 2003, I. P. Shestakov and U. U. Umirbaev [J. Am. Math. Soc. 17, No. 1, 181–196 (2004; Zbl 1044.17014); ibid. 17, No. 1, 197–227 (2004; Zbl 1056.14085)] proved that non-tame automorphisms exist and that the famous automorphism constructed by Nagata in 1972 is indeed non tame. In the paper under review, the author gives a reworked and clarified proof of this result.
The strategy is essentially along the same lines as in the original article of Shestakov and Umirbaev. However two new ingredients introduced by the author enable to shorten and simplify substantially the proof: a generalized notion of (multi)-degree leads to important simplifications in technical arguments and a notion of “weak Shestakov-Umirbaev reduction” is defined, which helps to clarify the previous notions of reductions of types I-IV introduced by Shestakov and Umirbaev. In particular, it is shown that the hypothetical type IV reduction needed in Shestakov and Umirbaev argument in fact never occurs (the similar question for types II and III is still open).
The paper is still quite long and technical but, in contrast with the original series of article by Shestakov and Umirbaev, it has the advantage to provide full, self-contained and detailed proofs of almost all intermediate lemmas. Also, the final criterion for non-tameness obtain by the author is much more flexible whence potentially applicable to a wider class of automorphisms.

MSC:

14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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References:

[1] A. van den Essen, L. Makar-Limanov and R. Willems, Remarks on Shestakov-Umirbaev, Report 0414, Radboud University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands, 2004.
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[9] I. Shestakov and U. Umirbaev, Poisson brackets and two-generated subalgebras of rings of polynomials, J. Amer. Math. Soc. 17 (2004), 181–196. · Zbl 1044.17014 · doi:10.1090/S0894-0347-03-00438-7
[10] I. Shestakov and U. Umirbaev, The tame and the wild automorphisms of polynomial rings in three variables, J. Amer. Math. Soc. 17 (2004), 197–227. · Zbl 1056.14085 · doi:10.1090/S0894-0347-03-00440-5
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