### Operators on generalized power series

Joris van der Hoeven
2001 Illinois Journal of Mathematics
Given a ring C and a totally (resp. partially) ordered set of "monomials" M, Hahn (resp. Higman) defined the set of power series 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 additional structure: if C is a field and M a totally ordered group, then Hahn proved that C[[M]] is a field. More recently, we have constructed fields of "transseries" of the form C[[M]] on which we defined natural derivations and
more » ... . 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 Φ: , 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. M , N , ) the orderings on such monomial sets. Usually, M is even a monomial monoid or group, on which the multiplication is assumed to be compatible with the ordering, i.e. m n ⇔ m v n v ⇔ v m v n, for all m, n, v ∈ M. Example 1. 1. M = {x α e βx |α, β ∈ R} with x α e βx 1 ⇔ (β > 0 ∨ (β = 0 ∧ α > 0)) is a totally ordered monomial group. Algebras of Noetherian series Assume now that C is a (not necessarily commutative) ring, and M a (not necessarily commutative) monomial monoid. Then we may naturally see C and M as subsets of C[[M]] via c c · 1 resp. m 1 · m. Given f and g in C[[M]], we define their product by f g = (m,n)∈supp f ×supp g f m g n m n. Noetherian series 5