On the complexity of solving ordinary differential equations in terms of Puiseux series

We prove that the binary complexity of solving ordinary polynomial differential equations in terms of Puiseux series is single exponential in the number of terms in the series. Such a bound was given by Grigoriev [10] for Riccatti differential polynomials associated to ordinary linear differential operators. In this paper, we get the same bound for arbitrary differential polynomials. The algorithm is based on a differential version of the Newton-Puiseux procedure for algebraic equations. Introduction In this paper we are interested in solving ordinary polynomial differential equations. For such equations we are looking to compute solutions in the set of formal power series and more generality in the set of Puiseux series. There is no algorithm which decide whether a polynomial differential equation has Puiseux series as solutions. We get an algorithm which computes a finite extension of the ground field which generates the coefficients of the solutions. Algorithms which estimate the coefficients of the solutions are given in [12, 13]. In order to analyse the binary complexity of factoring ordinary linear differential operators, Grigoriev [10] describes an algorithm which computes a fundamental system of solutions of the Riccatti equation associated to an ordinary linear differential operator. The binary complexity of this algorithm is single exponential in the order n of the linear differential operator. There are also algorithms for computing series solutions with real exponents [11, 1, 8, 2] and complex exponents [8]. Let K = Q(T1, . . . , Tl)[η] be a finite extension of a finitely generated field over Q. The variables T1, . . . , Tl are algebraically independent over Q and η is an algebraic element over the field Q(T1, . . . , Tl) with the minimal polynomial φ ∈ Z[T1, . . . , Tl][Z]. Let K be an algebraic closure of K and consider the two fields: L = ∪ν∈N∗K((x 1 ν )), L = ∪ν∈N∗K((x 1 ν )) which are the fields of fraction-power series of x over K (respectively K), i.e., the fields of Puiseux series of x with coefficients in K (respectively K). Each element ψ ∈ L (respectively ψ ∈ L) can be represented in the form ψ = ∑ i∈Q cix , ci ∈ K (respectively ci ∈ K). The