A note on isotropy groups and simple derivations

In the paper, we study the isotropy groups of K[x] if D is a simple derivation. We prove that Aut(K[x])(D) = {(x(1), ..., x(n-1), x(n) + c)vertical bar c is an element of K} if D = partial derivative(1) + (x(1)x(2) + 1)partial derivative(2) + x(2)partial derivative(3) or D = (1 - x(1)x(2))partial derivative(1) + x(1)(3)partial derivative(2) + x(2)partial derivative(3) + ... +x(n-1)partial derivative(n). We also prove that Aut(K[x])(D) = {id} if D = p(x(1))partial derivative(1) + Sigma(n)(i=2) q(i) (x(1), x(i))partial derivative(i), degx(2) q(2) = ... = deg(xn) q(n) and D is simple. Thus, we conjecture that Aut(K[x])(D) is conjugate to a subgroup of translations if D is simple. In addition, we prove some properties of simple derivations.