On Ritt's decomposition Theorem in the case of finite fields

TL;DR

This paper explores the generalization of Ritt's decomposition theorem for univariate polynomials over finite fields without the tameness condition, analyzing how decomposition chains behave in this broader context.

Contribution

It extends Ritt's classical theorem to finite fields without the tameness assumption, providing new insights into polynomial decomposition structures.

Findings

Decomposition chains may vary in length over finite fields without tameness.

The classical invariance of chain length does not hold universally in this setting.

New conditions or structures influencing decomposition are identified.

Abstract

A classical theorem by Ritt states that all the complete decomposition chains of a univariate polynomial satisfying a certain tameness condition have the same length. In this paper we present our conclusions about the generalization of these theorem in the case of finite coefficient fields when the tameness condition is dropped.