A commuting derivations theorem on UFD ’ s

Let A be the polynomial ring over k (a field of characteristic zero) in n + 1 variables. The commuting derivations conjecture states that n commuting locally nilpotent derivations on A, linearly independent over A, must satisfy A D 1 ,...,Dm = k[f ] where f is a coordinate. The conjecture can be formulated as stating that a (G m) n-action on k n+1 must have invariant ring k[f ] where f is a coordinate. In this paper we prove a statement (theorem 2.1) where we assume less on A (A is a UFD over k of transcendence degree n + 1 satisfying A * = k) and prove less (A/(f − α) is a polynomial ring for all but finitely many α). Under certain additional conditions (the D i are linearly independent modulo (f − α) for each α ∈ k) we prove that A is a polynomial ring itself and f is a coordinate. This statement is proven even more generally by replacing " free unipotent action of dimension n " for " G n a-action ". We make links with the (Abhyankar-)Sataye conjecture and give a new equivalent formulation of the Sataye conjecture.