On multidegree of tame and wild automorphisms of C^3

TL;DR

This paper investigates the multidegree sets of automorphisms of C^3, demonstrating the existence of wild automorphisms with multidegrees outside the tame set and characterizing conditions for tame multidegrees.

Contribution

It provides the first example of a wild automorphism with a multidegree not in the tame set and characterizes multidegrees for tame automorphisms when certain gcd conditions hold.

Findings

The set of multidegrees of automorphisms of C^3 intersects with that of tame automorphisms, and this intersection is infinite.

An explicit example of a wild automorphism with a multidegree outside the tame set is given.

For odd coprime degrees d_1 and d_2, the multidegree (d_1,d_2,d_3) belongs to tame automorphisms iff d_3 is a linear combination of d_1 and d_2.

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.