SIMPLE QUADRATIC DERIVATIONS IN TWO VARIABLES

Abstract

Let k[x, y] be the polynomial ring in two variables over an algebraically closed field k of characteristic zero. We call quadratic derivations the derivations of k[x, y] of the form

where a(x), b(x) ∈ k[x]. We are interested in simple derivations of this type; every such derivation is equivalent to Δ _p = ∂/∂x + (y ² − p(x))∂/∂y for a suitable p in k[x].

For some p, we are able to decide the simplicity of Δ _p : if the degree of p is odd, then Δ _p is simple; if p has degree 2, then Δ _p is simple if and only if p fulfills an arithmetic condition.

^*Supported by KBN Grant 2 PO3A 017 16.

Acknowledgments

Notes

^*Supported by KBN Grant 2 PO3A 017 16.