Abstract
Nagata (On Automorphism Group of k[x, y], Lectures in Mathematics, Department of Mathematics, Kyoto University, vol. 5. Kinokuniya Book-Store Co. Ltd., Tokyo, 1972) conjectured the wildness of a certain automorphism of the polynomial ring in three variables. This famous conjecture was solved by Shestakov–Umirbaev (J. Am. Math. Soc., 17, 181–196, 197–227, 2004) in the affirmative. Although the Shestakov–Umirbaev theory is powerful and applicable to various situations, not so many researchers seem familiar with this theory due to its technical difficulty.
In this paper, we explain how to prove the wildness of polynomial automorphisms using this theory practically. First, we recall a useful criterion for wildness which is derived from the generalized Shestakov–Umirbaev theory. Then, we demonstrate how to use this criterion effectively by showing the wildness of the exponential automorphisms for some well-known locally nilpotent derivations of rank three.
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Acknowledgements
The author is partly supported by the Grant-in-Aid for Young Scientists (B) 24740022, Japan Society for the Promotion of Science. He also thanks the referees for the comments.
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Kuroda, S. (2014). How to Prove the Wildness of Polynomial Automorphisms: An Example. In: Cheltsov, I., Ciliberto, C., Flenner, H., McKernan, J., Prokhorov, Y., Zaidenberg, M. (eds) Automorphisms in Birational and Affine Geometry. Springer Proceedings in Mathematics & Statistics, vol 79. Springer, Cham. https://doi.org/10.1007/978-3-319-05681-4_21
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DOI: https://doi.org/10.1007/978-3-319-05681-4_21
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