Tame automorphisms with multidegrees in the form of arithmetic progressions

Abstract

Let $(a,a+d,a+2d)$ be an arithmetic progression of positive integers. The following statements are proved: (1) If $a\mid 2d$, then $(a, a+d, a+2d)\in\mdeg(\Tame(\mathbb{C}^3))$. (2) If $a\nmid 2d$, then, except for arithmetic progressions of the form $(4i,4i+ij,4i+2ij)$ with $i,j \in\mathbb{N}$ and $j$ is an odd number, $(a, a+d, a+2d)\notin\mdeg(\Tame(\mathbb{C}^3))$. We also related the exceptional unknown case to a conjecture of Jie-tai Yu, which concerns with the lower bound of the degree of the Poisson bracket of two polynomials.