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 $x?x+a{x}^{q^s}+b{x}^{q^{n+s}}$ with $\gcd ?(n,s)=1$. It is known that the kernel of f has dimension at most 2, as proved by Csajbók et al. in [9]. For n big enough, e.g. $n\ge 5$ 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.