The inverse and the composition in the set of formal Laurent series

TL;DR

This paper explores the properties of multiplicative inverses and composition operations within formal Laurent series, highlighting conditions for invertibility and analyzing algebraic laws like distributivity and the chain rule.

Contribution

It provides necessary and sufficient conditions for the existence of inverses and introduces a general composition framework for formal Laurent series.

Findings

Inverses of formal Laurent series are not always unique.

Conditions for the existence of inverses are characterized.

The paper examines algebraic laws such as distributivity and the chain rule in this context.

Abstract

The aim of this article is to investigate the issues of multiplicative inverses and composition in the set of formal Laurent series. We show the lack of general uniqueness of inverses of formal Laurent series; necessary and sufficient conditions for the existence of inverses of formal Laurent series satisfying some weak assumptions are also provided. Moreover, we define a general composition of formal Laurent series and investigate the Right Distributive Law and the Chain Rule in this context.