Abstract

Let f be a function from N to N that can not be computed in polynomial time, and let a be an element of a differential field K of characteristic 0. The problem of large powers is the set of tuples $\bar{x} = $ of K so that $x_1 = a^{f}$, and the problem of large roots is the set of tuples $\bar{x}$ of K so that $x^{f}_1 = a$. These are two examples of problems that the use of derivation does not allow to solve quicker. We show that the problem of large roots is not polynomial for the differential field K, even if we use a polynomial number of parameters, and that the problem of large powers is not polynomial for the differential field K, even for non-uniform complexity. The proofs use the polynomial stability 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 $\bar{x}$ into a polynomial in variables $\bar{x}$. and their derivatives.