Computing the Lie Algebra of the Differential Galois Group of a Linear Differential System Moulay Barkatou, Thomas Cluzeau, Jacques-Arthur Weil Univ. de Limoges, CNRS ; XLIM UMR 7252 123 av. Albert Thomas, 87 060 Limoges, France forename.surname@unilim.fr Lucia Di Vizio Univ. de Versailles Saint-Quentin-En-Yvelines Laboratoire de Mathématiques ; UMR 8100 45 av. des États-Unis, 78035 Versailles, France divizio@math.cnrs.fr ABSTRACT We consider a linear differential system [A]: y ′ = A y with coefficients in C(x). The differential Galois group G of [A] is a linear algebraic group which measures the algebraic re- lations among solutions. Although there exist general al- gorithms 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. Keywords Computer algebra; Algorithms; Linear differential systems; Differential Galois theory; Lie algebras; Grothendieck-Katz p-curvature conjecture; Eigenrings; Reduced forms 1. INTRODUCTION Consider a linear differential system [A]: y ′ = A y with A ∈ Mn(C(x)) and n> 1. Its differential Galois group G measures everything that algebra can tell about the solu- tions, see [25]. For example, it measures solvability (with applications to integrability of dynamical systems, see ref- erences in [3, 1]), reducibility, transcendance properties for number theory, and so on. In theory, there exist general algorithms for computing differential Galois groups. Com- point and Singer gave such an algorithm in [12] in the case of reductive groups. Hrushovski gave a general algorithm in [19] which was recently clarified and improved by Feng in [16]. A symbolic-numeric algorithm is proposed by van der Hoeven in [24], based on the Schlesinger-Ramis density the- orems. However, although these are wonderful decision pro- cedures, none of them are either practical or implemented. For a large class of problems, it is sufficient to compute the Lie algebra g of G (which amounts to computing the connected component of the identity G ◦ ) instead of the dif- ferential Galois group G itself. See, for instance, Morales- Ramis-Sim´ o integrability theory ([3]) or the work by Nguyen and van der Put in [22] where they study when a given dif- Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full cita- tion on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or re- publish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. ISSAC ’16, July 19-22, 2016, Waterloo, ON, Canada c  2016 ACM. ISBN 978-1-4503-4380-0/16/07. . . $15.00 DOI: http://dx.doi.org/10.1145/2930889.2930932 ferential system can be solved in terms of systems of lower order. The purpose of the present paper is to use a similar philosophy for computing g. Our starting point is the theory of Katz ([21]). Let M be the differential module associated with [A]. There is a theoretical identification (tannakian correspondence) between g and a submodule W of End(M). Our main contribution is to make this identification algorith- mic and provide an effective algorithm to compute g when M is absolutely irreducible. To achieve this, we proceed in four main steps. The first step (Section 3) consists in computing a maximal decompo- sition of End(M). Using eigenring techniques, this requires to compute rational solutions ([4]) of a structured system of dimension n 4 . By exploiting the structure of the sys- tem, we reduce this problem to computing rational solutions of systems of lower dimensions which significantly improves the complexity of this step. In Section 4, to find a candi- date for the submodule W (corresponding to g), we use a modular approach based on Grothendieck-Katz conjecture (see Conjecture 4.1). We choose a prime p, compute the p- curvature χp ([10]) and identify the smallest submodule of End(M) whose reduction modulo p contains χp. This pro- vides a guess for W which is given by a basis M1,...,M d of matrices in Mn(C(x)). The next steps (Section 5) rely on the fact that g can also be directly read off from a reduced form of [A] (see Theorem 5.1). Using recent results from [2], we then prove that computing a reduction matrix amounts to computing a conjugation matrix between two semisim- ple Lie subalgebras of gln(C(x)) respectively generated by the Mi and their evaluations Mi (x0) at some ordinary point x0 of [A]. For the third step of our algorithm, we use a method for computing conjugation matrices based on re- sults on semisimple Lie algebras ([20, 14]). In our last step, we find a reduction matrix among the conjugation matrices. If our guess for W is not correct, then the third and fourth steps may fail. In this case, we go back to the second step and restart with another prime. The resulting algorithm is deterministic, assuming the ve- racity of the Grothendieck-Katz conjecture. We also give a less fancy algorithm which works independently of the Grothendieck-Katz conjecture. Note that a reduced form is obtained as a byproduct of our algorithm. We have a prototype implementation of our algorithm in Maple. We have applied it to many examples and it turns out that the most costly step is the decomposition of End(M). This step has a polynomial complexity in n ([7]) compared to the exponential (several levels) complexity in n of the existing algorithms for computing G ([16]).