Factorization of a class of composed polynomials

TL;DR

This paper analyzes the factorization of composed polynomials over finite fields, providing degree distributions of irreducible factors, applications for constructing irreducible polynomials, and explicit factorizations under specific conditions.

Contribution

It introduces a detailed degree distribution of irreducible factors for composed polynomials involving linearized polynomials over finite fields, with practical applications.

Findings

Degree distribution of irreducible factors of $f(L(x))$ over $\\mathbb{F}_q$

Lower bounds for the number of irreducible factors

Explicit factorization of $f(x^q-x)$ under certain conditions

Abstract

In this paper, we provide the degree distribution of irreducible factors of the composed polynomial $f(L(x))$ over $\mathbb F_q$, where $f(x)\in \mathbb F_q[x]$ is irreducible and $L(x)\in \mathbb F_q[x]$ is a linearized polynomial. We further provide some applications of our main result, including lower bounds for the number of irreducible factors of $f(L(x))$, constructions of high degree irreducible polynomials and the explicit factorization of $f(x^q-x)$ under certain conditions on $f(x)$.