Normal form for Ritt's Second Theorem

TL;DR

This paper introduces a normal form for Ritt's Second Theorem on polynomial decompositions, simplifying its application and clarifying the relationship between different decomposition types, especially over finite fields.

Contribution

It provides a new normal form for Ritt's Second Theorem, making the theorem easier to apply and understand, and offers an exact count of collisions in the tame case.

Findings

Normal form simplifies Ritt's Second Theorem application

Clarifies the relation between decomposition types

Counts collisions in the tame case

Abstract

Ritt's Second Theorem deals with composition collisions g o h = g* o h* of univariate polynomials over a field, where deg g = deg h*. Joseph Fels Ritt (1922) presented two types of such decompositions. His main result here is that these comprise all possibilities, up to some linear transformations. Because of these transformations, the result has been called "difficult to use". We present a normal form for Ritt's Second Theorem, which is hopefully "easy to use", and clarify the relation between the two types of examples. This yields an exact count of the number of such collisions in the "tame case", where the characteristic of the (finite) ground field does not divide the degree of the composed polynomial.