Tame automorphisms of C^3 with multidegree of the form (p_1,p_2,d_3)

TL;DR

This paper characterizes when a sequence of integers can be the multidegree of tame automorphisms of complex three-dimensional space, linking it to linear combinations of prime numbers.

Contribution

It provides a complete characterization of multidegrees (p_1,p_2,d_3) for tame automorphisms of C^3 based on linear combinations of primes.

Findings

Multidegree (p_1,p_2,d_3) corresponds to tame automorphisms if and only if d_3 is in p_1*N + p_2*N.

D_3 must be a linear combination of p_1 and p_2 with non-negative integer coefficients.

The result applies when p_1,p_2 are prime and p_2 > p_1 >= 3.

Abstract

Let d_3 >= p_2 > p_1 >= 3 be integers such that p_1,p_2 are prime numbers. In this paper we show that the sequence (p_1,p_2,d_3) is the multidegree of some tame automorphisms of C^3 if and only if d_3 is in p_1*N+p_2*N, i.e. if and only if d_3 is a linear combination of p_1 and p_2 with coefficients in N.