On an application of L. Ehrenpreis' method to ordinary differential equations

In 1956 Ehrenpreis [3] considered an application of the sheaf theory to differential equations and gave a criterion for the existence of global solutions of differential equations where the existence of local solutions are assured. We shall apply this method to systems of ordinary and linear differential equations with coefficients meromorphic in a domain D on the plane C of one complex variable z. Let D and Wl be the sheaves of all germs of functions holomorphic and meromorphic in D respectively. Let a^ (j, k=l, 2, •••,/>) be functions meromorphic in D. For any element f=(f,f, , f p ) of 30̂ , we define