The problem of large powers and the problem of large roots

Let f he a function from N to N that can not be computed in polynomial time . and let a be an element of a differential held K of characteristic 0. The problem of large powers is the set of tuples (x) over bar = (x(1).....x(n) ) of K so that x(1) = a(t(n)) and the problem of large roots in the set of tuples (x) over bar of K so that x(1)(t(n)) - a. These are two tramples of problems that the use of derivation does not allow to solve quicker. We sho w that the problem of large roots is not polynomial for the differential fi eld K. even if we use a polynomial number of parameters and that the proble m of large powers is not polynomial for the differential field K. even for nun-uniform complexity The proofs use thr polynomial stability (i.e.. the e limination of parameters) of field of characteristic 0. shown by L. Blum. F . Cucker. M. Shub and S. Smale. and the reduction lemma. that transforms a differential polynomial in variables (v) over bar into a polynomial in vari ables and their derivatives.