On Ritt's polynomial decomposition theorems

TL;DR

This paper advances the understanding of polynomial decomposition by providing a new description of all decompositions, addressing limitations in Ritt's original procedure, with applications in dynamical systems and invariant curves.

Contribution

It introduces a novel description of polynomial decompositions that controls the number of transformations needed, improving upon Ritt's classical results.

Findings

New description of polynomial decompositions

Applications to polynomial orbit intersections

Applications to invariant affine curves

Abstract

Ritt studied the functional decomposition of a univariate complex polynomial f into prime (indecomposable) polynomials, f = u_1 o u_2 o ... o u_r. His main achievement was a procedure for obtaining any decomposition of f from any other by repeatedly applying certain transformations. However, Ritt's results provide no control on the number of times one must apply the basic transformations, which makes his procedure unsuitable for many theoretical and algorithmic applications. We solve this problem by giving a new description of the collection of all decompositions of a polynomial. Our results have been used by Ghioca, Tucker and Zieve (arXiv:0807.3576) to describe the polynomials f,g having orbits with infinite intersection; they have also been used by Medvedev and Scanlon to describe the affine curves invariant under a coordinatewise polynomial action.