We give a new algorithm for the formal reduction of linear differential systems with Laurent series coefficients. We show how to obtain a decomposition of Balser, Jurkat and Lutz using eigenring techniques. We establish structural information on the obtained indecomposable subsystems and retrieve information on their invariants such as ramification. We show why classical algorithms then perform well on these subsystems. We also give precise estimates of the precision on the power series which is required in each step of our algorithm. The algorithm is implemented in Maple. We give examples in [14].

We consider a linear differential system [A] : y'=A, y}, where A has with coefficients in C(x). The differential Galois group G of [A] is a linear algebraic group which measures the algebraic relations among solutions. Although there exist general algorithms to compute $G$, none of them is either practical or implemented. This paper proposes an algorithm to compute the Lie algebra g of G when [A] is absolutely irreducible. The algorithm is implemented in Maple.

We present algorithms for computing rational and hyperexponential solutions of linear D-finite partial differential systems written as integrable connections. We show that these types of solutions can be computed recursively by adapting existing algorithms handling ordinary linear differential systems. We provide an arithmetic complexity analysis of the algorithms that we develop. A Maple implementation is available and some examples and applications are given.

We consider an analytic vector field x = X(x\right) and study, via a variational approach, whether it may possess analytic first integrals. We assume one solution Γ is known and we study the successive variational equations along Γ. Constructions in [MRRS07] show that Taylor expansion coefficients of first integrals appear as rational solutions of the dual linearized variational equations. We show that they also satisfy linear "filter" conditions. Using this, we adapt the algorithms from [Bar99, vHW97] to design new ones optimized to this effect and demonstrate their use. Part of this work stems from the first author's Ph.D. thesis1 [AM10].

Given a second order linear differential equations with coefficients in a field k=C(x), the Kovacic algorithm finds all Liouvillian solutions, that is, solutions that one can write in terms of exponentials, logarithms, integration symbols, algebraic extensions, and combinations thereof. A theorem of Klein states that, in the most interesting cases of the Kovacic algorithm (i.e when the projective differential Galois group is finite), the differential equation must be a pullback (a change of variable) of a standard hypergeometric equation. This provides a way to represent solutions of the differential equation in a more compact way than the format provided by the Kovacic algorithm. Formulas to make Klein's theorem effective were given in [4, 2, 3]. In this paper we will give a simple algorithm based on such formulas. To make the algorithm more easy to implement for various differential fields k, we will give a variation on the earlier formulas, namely we will base the formulas on invariants of the differential Galois group instead of semi-invariants.

We present alternative algorithms for computing symmetric powers of linear ordinary differential operators.Our algorithms are applicable to operators with coefficients in arbitrary integral domains and become faster than the traditional methods for symmetric powers of sufficiently large order, or over sufficiently complicated coefficient domains.The basic ideaa are also applicable to other computations involving cyclic vector techniques, such as exterior powers of differential or difference operators.

In general, there is no method for finding closed form first integrals or solutions of ordinary differential equations with non-constant coefficients. Thus, one usually performs heuristics, but this involves fastidious computations. The aim of this paper is to propose strategies that computerize such heuristics to help the analysis.