Roots of certain polynomials over finite fields

Abstract

We determine the roots in Fq3 of the polynomial X2qℓ+1+X+a for each positive integer ℓ and each a∈Fq, where q is a power of 2. We introduce a new approach for this type of question, and we obtain results which are more explicit than the previous results in this area. Our results resolve an open problem and a conjecture of Zheng, Kan, Zhang, Peng, and Li.