Formal solutions of linear differential equations
نویسنده
چکیده
A new approach to formal solutions of (first order systems of) linear differential equations stressing as much as possible the similarity with systems of linear equations. This approach leads to a fine structure of formal solutions, comparable to [1] but from a different point of view. A quick introduction to characteristic classes (cf. [3]) will be given. 1 Linear algebra; recalling Jordan decomposition Notations • K: field, perfect in Theorems 1 and 2 below. • K̄: algebraic closure of K • Mm,n(K): matrices having m rows, n columns and entries in K • Mn(K) =Mn,n(K) • V : K-vector space, dimK(V ) <∞ Theorem 1 (Abstract Jordan decomposition) For f ∈ EndK(V ) there exist fs, fn ∈ EndK(V ) such that • fs is semi-simple, fn nilpotent. • f = fs + fn. • [fs, fn] def = fs ◦ fn − fn ◦ fs = 0. fs, fn are uniquely determined by these conditions. Additional property: There exists P ∈ K[T ] such that P has no constant term and fs = P (f). (Similar for fn.)
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تاریخ انتشار 2002