A complete presentation of $(\mathbb{C}\mbox{[X]},\circ)$

TL;DR

This paper provides a comprehensive overview of the monoid of complex polynomials under composition, including fundamental theorems, canonical forms, and a detailed list of decomposable polynomials up to degree 12.

Contribution

It offers a complete presentation of the monoid of complex polynomials under composition, extending classical results with explicit classifications.

Findings

Recalls the fundamental theorem of Ritt and analyzes affine group actions.

Provides canonical forms for complex polynomials.

Lists decomposable polynomials up to degree 12.

Abstract

We recall the fundamental theorem of J.F. Ritt, with a stress on the action of the affine group and canonical forms of complex polynomials. Then we give a complete presentation of the monoid $(\mathbb{C}\mbox{[X]},\circ)$. A list of decomposable polynomials is given in degrees $\leq 12$.