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

Let f be the Fq-linear map over Fqjavax.xml.bind.JAXBElement@3e829f17 defined by x↦x+axqjavax.xml.bind.JAXBElement@df90dc4+bxqjavax.xml.bind.JAXBElement@5c033a86 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≥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.