Factorisation of germ-like series

Abstract

A classical tool in the study of real closed fields are the fields K((G)) of generalised power series (i.e., formal sums with well-ordered support) with coefficients in a field K of characteristic 0 and exponents in an ordered abelian group G. A fundamental result in [1] ensures the existence of irreducible series in the subring K((G≤0)) of K((G)) consisting of the generalised power series with non-positive exponents.

It is an open question whether the factorisations of a series of the ring have common refinements, and whether the factorisation becomes unique after taking the quotient by the ideal generated by the non-constant monomials. In this paper, we provide a new class of irreducibles and prove some further cases of uniqueness of the factorisation.