On the asymptotic classification of Kloosterman polynomials

In2004,duringtheinvestigationofsomeidentitiesinvolvingKloostermansumsoverF2n,Hollmann and Xiang introduced the concept of Kloosterman polynomials. Let Tr2n (·) be the absolute trace map on F2n . For a subset Λ ⊆ {0,1··· ,n−1} with |Λ| even, they conjectured that the map fΛ : x 7→ x+Pi∈Λ x(2n−2)2i is a Kloosterman polynomial, which means that fΛ is injective on {x ∈ F2n : Tr2n (x) = 1}, if and only if Λ = {0, 1}, {1, 2} or {0, 3}. In the same paper, Hollmann and Xiang proved the sufficiency part of this conjecture. In this article, we provide a partial answer to this conjecture: if n ≥ 4(max(Λ)+1) and fΛ is a Kloosterman polynomial, then Λ = {0,1}, {1,2} or {0,3}. To prove this result, we first transform this conjecture into a permutation polynomial problem. Then we use approaches and tools from function field theory to obtain a classification result on these polynomials.