On multidegrees of tame and wild automorphisms of C3

In this note we show that the set is not empty, where denotes multidegree. Moreover we show that this set has infinitely many elements. Since for Nagata’s famous example N of a wild automorphism, , and since for other known examples of wild automorphisms the multidegree is of the form (1,d2,d3) (after permutation if necessary), we give the very first example of a wild automorphism F of C3 with .We also show that, if d1,d2 are odd numbers such that gcd(d1,d2)=1, then if and only if d3∈d1N+d2N. This a crucial fact that we use in the proof of the main result.