Palindromic Polynomials over Finite Fields

TL;DR

This paper counts monic polynomials with specific palindromic factors over finite fields and applies the results to systems of linear equations of index 2, advancing understanding of polynomial structures in finite fields.

Contribution

It introduces a method to count polynomials with self-reciprocal factors over finite fields and applies this to analyze certain linear systems.

Findings

Derived formulas for counting polynomials with palindromic factors

Established connections between polynomial factorization and linear systems

Provided applications to systems of linear equations of index 2

Abstract

For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application is given for systems of linear equations over $\mathbb{F}$ of index $2$.