A note on the computation of Puiseux series solutions of the Riccatti equation associated with a homogeneous linear ordinary differential equation

We present in this paper a detailed note on the computation of Puiseux series solutions of the Riccatti equation associated with a homogeneous linear ordinary differential equation. This paper is a continuation of [1] which was on the complexity of solving arbitrary ordinary polynomial differential equations in terms of Puiseux series. Introduction LetK = 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]. LetK 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 order of ψ is defined by ord(ψ) := min{i ∈ Q, ci 6= 0}. The fields L and L are differential fields with the differentiation d dx (ψ) = ∑