Extending a result of Carlitz and McConnel to polynomials which are not permutations

Let D denote the set of directions determined by the graph of a polynomial f of F q [ x ] , where q is a power of the prime p . If D is contained in a multiplicative subgroup M of F q × , then by a result of Carlitz and McConnel it follows that f ( x ) = a x p k + b for some k ∈ N . Of course, if D ⊆ M , then 0 ∉ D and hence f is a permutation. If we assume the weaker condition D ⊆ M ∪ { 0 } , then f is not necessarily a permutation, but Sziklai conjectured that f ( x ) = a x p k + b follows also in this case. When q is odd, and the index of M is even, then a result of Ball, Blokhuis, Brouwer, Storme and Szőnyi combined with a result of Göloğlu and McGuire proves the conjecture. Assume deg ⁡ f ≥ 1 . We prove that if the size of D − 1 D = { d − 1 d ′ : d ∈ D ∖ { 0 } , d ′ ∈ D } is less than q − deg ⁡ f + 2 , then f is a permutation of F q . We use this result to prove the conjecture of Sziklai.