Simple derivations in two variables

Simple derivations and their images

“…" ⇐ " Let a 0 := a 0 (x) ∈ K * . If a 2 (x) ∈ K, then it follows from Theorem 2.2 in [12] that D is simple. If deg a 2 (x) ≥ 1, a 1 ∈ K, then it follows from Theorem 4.1 that D is simple.…”

“…, y n }. For some examples of simple derivations, see [1], [6], [7], [9], [12]. A polynomial F ∈ K[x, y] is said to be a Darboux polynomial of D if F / ∈ K and D(F ) = ΛF for some Λ ∈ K[x, y].…”

“…In [5], S. Kour has shown that D = y r ∂ x + (xy s + g)∂ y , where 0 ≤ r < s are integers, g ∈ K[y], is a simple derivation of K[x, y]. In [12], the second author proved that D = y∂ x + (a 2 y 2 + a 1 (x)y + a 0 (x))∂ y with a 2 ∈ K, a 1 (x), a 0 (x) ∈ K[x] is simple iff a 0 (x) ∈ K * and deg a 1 (x) ≥ 1.…”

The images of some simple derivations

“…An ideal I of R is called D-stable if D(I) ⊂ I. R is called D-simple if it has no proper nonzero D-stable ideal. The K-derivation D is called simple if R has no D-stable ideals other than 0 and R. For some examples of simple derivations, see [1], [2], [3], [8].…”

On Shamsuddin derivations and the isotropy groups

“…An ideal I of R is called D-stable if D(I) ⊂ I. R is called D-simple if it has no proper nonzero D-stable ideal. The K-derivation D is called simple if R has no D-stable ideals other than 0 and R. For some examples of simple derivations, see [1], [3], [5], [6], [8], [9], [10], [11].…”