Functional equations in formal power series

TL;DR

This paper investigates functional equations within the semigroup of formal power series under composition over an algebraically closed field, providing new results on decompositions and structure of subsemigroups.

Contribution

It introduces novel results on equations in the semigroup of formal power series and addresses open questions about decompositions and conjugacy of subsemigroups.

Findings

Answered a question on decompositions of even formal power series.

Proved that every right amenable subsemigroup is conjugate to a subsemigroup of monomials.

Established general results about equations in the semigroup of formal power series.

Abstract

Let $k$ be an algebraically closed field of characteristic zero, and $k[[z]]$ the ring of formal power series over $k$. In this paper, we study equations in the semigroup $z^2k[[z]]$ with the semigroup operation being composition. We prove a number of general results about such equations and provide some applications. In particular, we answer a question of Horwitz and Rubel about decompositions of ``even'' formal power series. We also show that every right amenable subsemigroup of $z^2k[[z]]$ is conjugate to a subsemigroup of the semigroup of monomials.