On the integrability of polynomial fields in the plane by means of Picard-Vessiot theory

We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Li\'enard equations and equations related with special functions such as Hypergeometric and Heun ones. We also study the Poincar\'e problem for some of the families.Preprin

Non-integrability of some hamiltonians with rational potentials

In this paper we give a mechanism to compute the families of classical hamiltonians of two degrees of freedom with an invariant plane and normal variational equations of Hill-Schr\"odinger type selected in a suitable way. In particular we deeply study the case of these equations with polynomial or trigonometrical potentials, analyzing their integrability in the Picard-Vessiot sense using Kovacic's algorithm and introducing an algebraic method (algebrization) that transforms equations with transcendental coefficients in equations with rational coefficients without changing the Galoisian structure of the equation. We compute all Galois groups of Hill-Schr\"odinger type equations with polynomial and trigonometric (Mathieu equation) potentials, obtaining Galoisian obstructions to integrability of hamiltonian systems by means of meromorphic or rational first integrals via Morales-Ramis theory.Peer Reviewe

On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory

We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Liénard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincaré problem for some families is also approached

Differential equations of order two with one singular point

The goal of this paper is to describe the set of polynomials r is an element of C[x] such that the linear differential equation y " = ry has Liouvillian solutions, where C is an algebraically closed field of characteristic 0. It is known that the differential equation has Liouvillian solutions only if the degree of r is even. Using differential Galois theory we show that the set of such polynomials of degree 2n can be represented by a countable set of algebraic varieties of dimension n + 1. Some properties of those algebraic varieties are proved. (C) 1999 Academic Press