Relation between the number of roots of a linearized polynomial and projective polynomial.
If we have a linearized polynomial of $\sigma$-degree $2$ over $F_{q^n}$ where $\sigma=q^s$, we can factor out $x$ as follows:
$$l(x)=x^{\sigma^2}+ax^{\sigma}+bx=x(x^{\sigma^2-1}+ax^{\sigma-1}+b)=xl_1(x)$$.
then if $l(x)=0$, $x\neq 0$ and $y=x^{\sigma-1}$ we get
$$y^{\sigma+1}+ay+b=0$$
It is clear that $x=0$ is always a root for $l(x)=0$.
Question:
What is the relation between the number of roots of $l_1(x)=0$ over $F_{q^n}$ and the roots of the projective polynomial equation $p(y)=y^{\sigma+1}+ay+b=0$?