Abstract
We give a simpler proof as well as a generalization of the main result of an article of Shestakov and Umirbaev. This latter article being the first of two that solve a long-standing conjecture about the non-tameness, or “wildness”, of Nagata’s automorphism. As corollaries we get interesting informations about the leading terms of polynomials forming an automorphism of K[x 1, . . . , x n ] and reprove the tameness of automorphisms of K[x 1, x 2].
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References
Shestakov I.P., Umirbaev U.U.: Poisson brackets and two-generated subalgebras of rings of polynomials. J. Am. Math. Soc. 17, 181–196 (2004)
Shestakov I.P., Umirbaev U.U.: The tame and the wild automorphisms of polynomial rings in three variables. J. Am. Math. Soc. 17, 197–227 (2004)
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Vénéreau, S. A parachute for the degree of a polynomial in algebraically independent ones. Math. Ann. 349, 589–597 (2011). https://doi.org/10.1007/s00208-010-0531-5
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DOI: https://doi.org/10.1007/s00208-010-0531-5