具有一个素数的Tame自同构的多阶性
Multidegrees of Tame automorphisms with one prime number
摘要：
令$3\leq d_1\leq d_2\leq d_3$为整数。我们展示了以下结果：（1）若$d_2$是素数且$\frac{d_1}{\gcd(d_1,d_3)}\neq2$，则$(d_1,d_2,d_3)$是驯服自同构的多重次数当且仅当$d_1=d_2$或$d_3\in d_1\mathbb{N}+d_2\mathbb{N}$ ;（2）若$d_3$是素数且$\gcd(d_1,d_2)=1$，则$(d_1,d_2,d_3)$是驯服自同构的多重次数当且仅当$d_3\in d_1\mathbb{N}+d_2\mathbb{N}$ .我们还将这一调查与Drensky和Yu的一个猜想，这涉及到两个多项式的泊松括号的次数的下界，我们给这个猜想的反例。
英文摘要：
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.
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