Galois Action on Solutions of a Differential Equation

Consider a second-order differential equation of the form y″ + ay ′ + by = 0 with a, b ε Q(x). Kovacicʹs algorithm tries to compute a solution of the associated Riccati equation that is algebraic and of minimal degree over (x). The coefficients of the monic irreducible polynomial of this solution are in C(x), where C is a finite algebraic extension of Q. In this paper we give a bound for the degree of the extension C Q. Similar results are obtained for third-order differential equations.