Multidegrees of Tame automorphisms with one prime number

Abstract

Let $3\leq d_1\leq d_2\leq d_3$ be integers. We show the following results: (1) If $d_2$ is a prime number and $\frac{d_1}{\gcd(d_1,d_3)}\neq2$, then $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism if and only if $d_1=d_2$ or $d_3\in d_1\mathbb{N}+d_2\mathbb{N}$; (2) If $d_3$ is a prime number and $\gcd(d_1,d_2)=1$, then $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism if and only if $d_3\in d_1\mathbb{N}+d_2\mathbb{N}$. We also relate this investigation with a conjecture of Drensky and Yu, which concerns with the lower bound of the degree of the Poisson bracket of two polynomials, and we give a counter-example to this conjecture.