Generalized power series solutions to linear partial differential equations

Let Theta = C vertical bar e(-x1),...,e(-xn)parallel to partial derivative(1),...,partial derivative(n)vertical bar and S = C[x(1),...,x(n)parallel to[e(Cx1+...+Cxn)]], where C is an effective field and x(1)(N)...x(n)(N)e(cx1+...+Cxn) and S are given a suitable asymptotic ordering <=. Consider the mapping L : S ---> S-l; f --> (L(l)f,...,L(l)f), where L-l,...,L-l is an element of Theta. For g = (g(1),...,g(l)) is an element of S-L(l) = im L, it is natural to ask how to solve the system Lf = g. In this paper, we will effectively characterize S-L(l) and show how to compute a so called distinguished right inverse L-1 : S-L(l) --> S of L. We will also characterize the solution space of the homogeneous equation Lh = 0. (C) 2007 Published by Elsevier Ltd.