Removing Apparent Singularities of Systems of Linear Differential Equations with Rational Function Coefficients

International audienceLet (S) Y'=A(x)Y be a system of first order linear differential equations with rational function coefficients. A singular point x0 of (S) is called an apparent singularity if there is a basis of formal solutions of (S) which are holomorphic in a neighborhood of x0. In this talk we shall present a new algorithm which, given a system of the form (S), detects apparent singularities and constructs an equivalent system (S') with rational coefficients, such that every singularity of (S') is a singularity of (S) that is not apparent. Our method can, in particular, be applied to the companion system of any linear differential equation with arbitrary order n . We thus have an alternative method to the standard methods for removing apparent singularities of linear differential operators. We shall compare our method to the one designed for operators and we shall show some applications and examples of computation

Removing Apparent Singularities of Systems of Linear Differential Equations with Rational Function Coefficients

International audienceIn this paper we present a new algorithm which, given a system of first order linear differential equations with rational function coefficients, constructs an equivalent system with rational function coefficients, whose finite singularities are exactly the non-apparent singularities of the original system. This algorithm is implemented in the computer algebra system Maple and is illustrated by examples