JORIS VAN DER HOEVENAbstract. Given a ring Cand a totally (resp. partially) ordered setof \monomials" M, Hahn (resp. Higman) de ned the set of power se-ries C[[M]] with well-ordered (resp. Noetherian or well-quasi-ordered)support in M. This set C[[M]] can usually be given a lot of additionalstructure: if C is a eld and M a totally ordered group, then Hahnproved that C[[M]] is a eld. More recently, we have constructed eldsof \transseries" of the form C[[M]] on which we de ned natural deriva-tions and compositions.In this paper we develop an operator theory for generalized powerseries of the above form. We rst study linear and multilinear oper-ators. We next isolate a big class of so-called Noetherian operators : C[[M]] !C[[N]], which include (when de ned) summation, multi-plication, di erentiation, composition, etc. Our main result is the proofof an implicit function theorem for Noetherian operators. This theoremmay be used to explicitly solve very general types of functional equationsin generalized power series.
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