Asymptotic expansion of singular solutions and the characteristic polygon of linear partial differential equations in the complex domain

Let P(z, partial derivative) be a linear partial differential operator with holomorphic coefficients in a neighborhood Omega of z = 0 in Cd+1. Consider the equation P(z, partial derivative )u(z) = f(z), where u(z) admits singularities on the surface K = {z(o) = 0} and f(z) has an asymptotic expansion of Gevrey type with respect to z(o) as z(o) --> 0. We study the possibility of asymptotic expansion of u(z). We define the characteristic polygon of P(z, partial derivative) with respect to K and characteristic indices. We discuss the behavior of u(z) in a neighborhood of K, by using these notions. The main result is a generalization of that in [6].