ON THE ASYMPTOTIC CLASSIFICATION OF KLOOSTERMAN POLYNOMIALS

In 2004, during the investigation of some identities involving Kloosterman sums over F-2(n) , Hollmann and Xiang introduced the concept of Kloosterman polynomials. Let Tr-2n (.) be the absolute trace map on F2n . For a subset A C_ {0, 1 . . ., n - 1} with A even, they conjectured that the map f Lambda : x -> x+ iE Lambda x((2n-2)2i) is a Kloosterman polynomial, which means that f Lambda is injective on {x E F-2n : Tr-2n (x) = 1}, if and only if A = {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(A) + 1) and f Lambda is a Kloosterman polynomial, then A = {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.