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

Csajbók, Bence (2025) Extending a result of Carlitz and McConnel to polynomials which are not permutations. FINITE FIELDS AND THEIR APPLICATIONS, 108. No. 102683. ISSN 1071-5797 (In Press)

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Abstract

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)=ax^(p^k)+b for some k \in N. Of course, if D\subseteq M, then 0\notin D and hence f is a permutation. If we assume the weaker condition D \subseteq M \cup {0}, then f is not necessarily a permutation, but Sziklai conjectured that f(x)=ax^(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 Göloglu and McGuire proves the conjecture. Assume deg f \geq 1. We prove that if the size of D^(-1)D={d^(-1)d' : d \in D \setminus {0} d'\in 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.

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