Operators on generalized power series

Abstract

Given a ring $C$ and a totally (resp. partially) ordered set of "monomials" $\mathfrak{M}$, Hahn (resp. Higman) defined the set of power series $C [[ \mathfrak{M} ]]$ with well-ordered (resp. Noetherian or well-quasi-ordered) support in $\mathfrak{M}$. This set $C [[ \mathfrak{M} ]]$ can usually be given a lot of additional structure: if $C$ is a field and $\mathfrak{M}$ a totally ordered group, then Hahn proved that $C [[ \mathfrak{M} ]]$ is a field. More recently, we have constructed fields of ``transseries'' of the form $C [[ \mathfrak{M} ]]$ on which we defined natural derivations and compositions. In this paper we develop an operator theory for generalized power series of the above form. We first study linear and multilinear operators. We next isolate a big class of so-called Noetherian operators $\Phi : C [[ \mathfrak{M} ]] \rightarrow C [[ \mathfrak{N}]]$, which include (when defined) summation, multiplication, differentiation, composition, etc. Our main result is the proof of an implicit function theorem for Noetherian operators. This theorem may be used to explicitly solve very general types of functional equations in generalized power series.