On multidegrees of tame and wild automorphisms of C3

AbstractIn this note we show that the set mdeg(Aut(C3))∖mdeg(Tame(C3)) 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)∈mdeg(Tame(C3)), 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 mdegF∉mdeg(Tame(C3)).We also show that, if d1,d2 are odd numbers such that gcd(d1,d2)=1, then (d1,d2,d3)∈mdeg(Tame(C3)) if and only if d3∈d1N+d2N. This a crucial fact that we use in the proof of the main result