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arXiv · 0911.5468

On multidegree of tame and wild automorphisms of C^3

Marek Karas·Jakub Zygadlo
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Abstract

In this note we show that the set mdeg(Aut(C^3)) mdeg(Tame(C^3)) is not empty. Moreover we show that this set has infinitely many elements. Since for the famous Nagata's example N of wild automorphism, mdeg N =(5,3,1) is an element of mdeg(Tame(C^3)) and since for other known examples of wild automorphisms the multidegree is of the form (1,d_2,d_3) (after permutation if neccesary), then we give the very first exmple of wild automorphism F of C^3 such that mdeg F does not belong to mdeg(Tame(C^3)). We also show that, if d_1,d_2 are odd numbers such that gcd (d_1,d_2) =1, then (d_1,d_2,d_3) belongs to mdeg(Tame(C^3)) if and only if d_3 is a linear combination of d_1,d_2 with natural coefficients. This a crucial fact that we use in the proof of the main result.

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Marek Karas, Jakub Zygadlo. 2009-11-29. On multidegree of tame and wild automorphisms of C^3. https://doi.org/10.1016/j.jpaa.2011.04.003

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