On the differential simplicity of polynomial rings. (Q1399178)

The aim of the paper is to present some new examples of rings with a derivation \(d\) having no nontrivial \(d\)-invariant ideals. Let \(\beta\in \mathbb Q[X,Y]\) be an irreducible homogeneous polynomial of degree \(n\geqslant 3\). Then \(\mathbb C[X,Y]\) is \(d\)-simple with respect to the derivation \[ D=\left[\beta (X+Y) +b\right] \dfrac{\partial}{\partial X} +\beta \dfrac{\partial}{\partial Y} \] for any nonzero \(b\in \mathbb Q\). The proof is based on some geometrical considerations.