On multidegrees of tame and wild automorphisms of C3

In this note we show that the set mdeg(Aut(C-3))\mdeg(Tame(C-3)) is not empty, where mdeg denotes multidegree. Moreover we show that this set has infinitely many elements. Since for Nagata's famous example N of a wild automorphism, mdegN = (5, 3, 1) epsilon 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 necessary), we give the very first example of a wild automorphism F of C-3 with mdegF is not an element of 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)) epsilon mdeg(Tame(C-3)) if and only if d(3) epsilon d(1)N d(2)N. This a crucial fact that we use in the proof of the main result. (C) 2011 Elsevier B.V. All rights reserved.