Factorisation of germ-like series

TL;DR

This paper investigates the factorization properties of generalized power series over real closed fields, introducing new irreducible series and exploring conditions for unique factorization in these series rings.

Contribution

It introduces a new class of irreducible series and advances understanding of factorization uniqueness in generalized power series rings.

Findings

Identified a new class of irreducible series.

Proved cases where factorization is unique.

Addressed open questions on common refinements of factorizations.

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 of Berarducci ensures the existence of irreducible series in the subring $K((G^{\leq 0}))$ of $K((G))$ consisting of the generalised power series with non-positive exponents. It is an open question whether the factorisations of a series in such subring 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.