  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].
