On the number of roots of some linearized polynomials

TL;DR

This paper investigates the roots of certain linearized polynomials over finite fields, providing bounds, characterizations, and explicit root-finding methods, with applications to linear sets in projective geometry.

Contribution

It introduces new bounds and characterizations for the roots of specialized linearized polynomials, using smaller matrices and explicit root-finding techniques, advancing understanding in finite field theory and geometry.

Findings

Bounds and characterizations for the number of roots.

A method to explicitly find roots via smaller matrices.

Applications to linear sets with small spectrum of weights.

Abstract

Linearized polynomials appear in many different contexts, such as rank metric codes, cryptography and linear sets, and the main issue regards the characterization of the number of roots from their coefficients. Results of this type have been already proved in [7,10,24]. In this paper we provide bounds and characterizations on the number of roots of linearized polynomials of this form \[ ax+b_0x^{\sigma}+b_1x^{\sigma q^n}+b_2x^{\sigma q^{2n}}+\ldots+b_{t-1}x^{\sigma q^{n(t-1)}} \in \mathbb{F}_{q^{nt}}[x], \] with $\sigma$ a generator of the Galois group $\mathrm{Gal}(\mathbb{F}_{q^n}\colon\mathbb{F}_q)$. Also, we characterize the number of roots of such polynomials directly from their coefficients, dealing with matrices which are much smaller than the relative Dickson matrices and the companion matrices used in the previous papers. Furthermore, we develop a method to find explicitly the roots of a such polynomial by finding the roots of a $q^n$-polynomial. Finally, as an applications of the above results, we present a family of linear sets of the projective line whose points have a small spectrum of possible weights, containing most of the known families of scattered linear sets. In particular, we carefully study the linear sets in $\mathrm{PG}(1,q^6)$ presented in [9].