Fast computation of power series solutions of systems of differential   equations
  We propose new algorithms for the computation of the first N terms of a vector (resp. a basis) of power series solutions of a linear system of differential equations at an ordinary point, using a number of arithmetic operations which is quasi-linear with respect to N. Similar results are also given in the non-linear case. This extends previous results obtained by Brent and Kung for scalar differential equations of order one and two. 
Alin Bostan (INRIA Rocquencourt), Fr\'ed\'eric Chyzak (INRIA   Rocquencourt), Fran\c{c}ois Ollivier (LIX), Bruno Salvy (INRI...