A Carlitz type result for linearized polynomials

TL;DR

This paper investigates conditions under which two $q$-polynomials over finite fields have identical image sets when divided by $x$, with applications to maximum scattered linear sets in projective geometry.

Contribution

It provides necessary and sufficient conditions for $q$-polynomials over $qn$ to have matching image sets of $f(x)/x$ and $g(x)/x$, extending Carlitz type results.

Findings

Characterization of $q$-polynomials with equal image sets for $n \\leq 5$

Application to maximum scattered linear sets in projective geometry

Extension of classical Carlitz results to linearized polynomials

Abstract

For an arbitrary $q$-polynomial $f$ over $\mathbb{F}_{q^n}$ we study the problem of finding those $q$-polynomials $g$ over $\mathbb{F}_{q^n}$ for which the image sets of $f(x)/x$ and $g(x)/x$ coincide. For $n\leq 5$ we provide sufficient and necessary conditions and then apply our result to study maximum scattered linear sets of $\mathrm{PG}(1,q^5)$.