Composition collisions and projective polynomials

TL;DR

This paper investigates the decomposition of polynomials over finite fields, especially those with degrees as powers of the characteristic p, addressing gaps in understanding of their compositional structure.

Contribution

It provides new insights into the structure of polynomial compositions over finite fields when degrees are divisible by the characteristic p, focusing on powers of p.

Findings

Characterization of polynomial decompositions with degrees as powers of p

Identification of structural properties in compositions over finite fields

Extension of existing decomposition theories to new polynomial classes

Abstract

The functional decomposition of polynomials has been a topic of great interest and importance in pure and computer algebra and their applications. The structure of compositions of (suitably normalized) polynomials f=g(h) over finite fields is well understood in many cases, but quite poorly when the degrees of both components are divisible by the characteristic p. This work investigates the decomposition of polynomials whose degree is a power of p.