On certain linearized polynomials with high degree and kernel of small dimension

Let $f$ be the $\mathbb{F}q$-linear map over $\mathbb{F}{q^{2n}}$ defined by $x\mapsto x+ax^{{qs}+bx{q{n+s}}$} with $\gcd(n,s)=1$. It is known that the kernel of $f$ has dimension at most $2$, as proved by Csajb\'ok et al. in "A new family of MRD-codes" (2018). For $n$ big enough, e.g. $n\geq5$ when $s=1$, we classify the values of $b/a$ such that the kernel of $f$ has dimension at most $1$. To this aim, we translate the problem into the study of some algebraic curves of small degree with respect to the degree of $f$; this allows to use intersection theory and function field theory together with the Hasse-Weil bound. Our result implies a non-scatteredness result for certain high degree scattered binomials, and the asymptotic classification of a family of rank metric codes.