A characterization of linearized polynomials with maximum kernel

TL;DR

This paper characterizes $q$-polynomials over finite fields that have the maximum number of roots, providing necessary and sufficient conditions, and explores their properties and classifications for specific degrees and field extensions.

Contribution

It introduces a complete characterization of linearized polynomials with maximum kernel and classifies such polynomials for degrees up to $q^{n-2}$ over finite fields.

Findings

Characterization of polynomials with maximum kernel

Complete classification for degrees up to $q^{n-2}$ for small $n$

Insights into the splitting fields of $q$-polynomials

Abstract

We provide sufficient and necessary conditions for the coefficients of a $q$-polynomial $f$ over $\mathbb{F}_{q^n}$ which ensure that the number of distinct roots of $f$ in $\mathbb{F}_{q^n}$ equals the degree of $f$. We say that these polynomials have maximum kernel. As an application we study in detail $q$-polynomials of degree $q^{n-2}$ over $\mathbb{F}_{q^n}$ which have maximum kernel and for $n\leq 6$ we list all $q$-polynomials with maximum kernel. We also obtain information on the splitting field of an arbitrary $q$-polynomial. Analogous results are proved for $q^s$-polynomials as well, where $\gcd(s,n)=1$.