Simple derivations in two variables

Abstract

Let $k$ be a field of characteristic zero. If $c_1,c_2\in k\setminus \{0\},s,t\ge 1$ and $u\ge 0$, then it is shown that the $k$-derivations ${\partial }_x+x^u(c_1{x}^t{y}^s+c_2){\partial }_y$ and ${\partial }_x+x^u(c_1{x}^t+c_2{y}^{s+1}){\partial }_y$ of $k[x,y]$ are simple. We also give a necessary and sufficient condition for the $k$-derivation $y^r{\partial }_x+(c_1{x}^{t_1}{y}^{s_1}+c_2{x}^{t_2}{y}^{s_2}){\partial }_y$, where $r,t_1,s_1,t_2,s_2\ge 0$ and $c_1,c_2\in k$, of $k[x,y]$ to be simple.