Simple derivations in two variables

Abstract

Let k be a field of characteristic zero. If c 1 , c 2 ∈ k ∖ { 0 } , s , t ≥ 1 and u ≥ 0 , then it is shown that the k-derivations ∂ x + x u ( c 1 x t y s + c 2 ) ∂ y and ∂ x + x u ( c 1 x t + c 2 y s + 1 ) ∂ y of k[x, y] are simple. We also give a necessary and sufficient condition for the k-derivation y r ∂ x + ( c 1 x t 1 y s 1 + c 2 x t 2 y s 2 ) ∂ y , where r , t 1 , s 1 , t 2 , s 2 ≥ 0 and c 1 , c 2 ∈ k , of k[x, y] to be simple.