Indices d'un operateur differentiel matriciel et applications

Abstract

In this paper, one determines the formal index and the polynomial index of a matrix linear differential operator P with coefficients in Mn(C[x]) and detAm(x) not identically zero. Then, one applies these results to give a new proof of a Bezivin-Robba theorem equivalent to the Lindemann-Wierstrass theorem, as to find sufficient conditions on the Riccati matrix differential equation Y' = A(x)+B(x)Y +Y C(x)Y with coefficients in Mn(C[x]) so that any meromorphic solution is rational and other sufficient conditions so that the general solution is algebraic.