A reduced form for linear differential systems and its application to integrability of Hamiltonian systems

Let k be a differential field with algebraic closure , and let with A∈Mn(k) be a linear differential system. Denote by g the Lie algebra of the differential Galois group of [A]. We say that a matrix is a reduced form of [A] if and there exists such that . Such a form is often the sparsest possible attainable through gauge transformations without introducing new transcendents. In this paper, we discuss how to compute reduced forms of some symplectic differential systems, arising as variational equations of Hamiltonian systems. We use this to give an effective form of the Morales–Ramis theorem on (non-)-integrability of Hamiltonian systems.