We study a monomial derivation $d$ proposed by J. Moulin Ollagnier and A. Nowicki in the polynomial ring of four variables, and prove that $d$ has no Darboux polynomials if and only if $d$ has a trivial field of constants.
Let $3leq d_1leq d_2leq d_3$ be integers. We show the following results: (1) If $d_2$ is a prime number and $frac{d_1}{gcd(d_1,d_3)} eq2$, then $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism if and only if $d_1=d_2$ or $d_3in d_1mathbb{N}+d_
2mathbb{N}$; (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 $d_3in d_1mathbb{N}+d_2mathbb{N}$. 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.
Let $(a,a+d,a+2d)$ be an arithmetic progression of positive integers. The following statements are proved: (1) If $amid 2d$, then $(a, a+d, a+2d)inmdeg(Tame(mathbb{C}^3))$. (2) If $a mid 2d$, then, except for arithmetic progressions of the form $
(4i,4i+ij,4i+2ij)$ with $i,j inmathbb{N}$ and $j$ is an odd number, $(a, a+d, a+2d) otinmdeg(Tame(mathbb{C}^3))$. 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.
