Abstract

We call a pair of polynomials f,g in F_q[T] a Davenport pair (DP) if their value sets are equal, V_f(F_q^t) = V_g(F_q^t), for infinitely many extensions of F_q. If they are equal for all extensions of F_q (for all t ≥ 1), then we say (f,g) is a strong Davenport pair (SDP). Exceptional polynomials and SDP’s are special cases of DP’s. Monodromy/Galois-theoretic methods have successfully given much information on exceptional polynomials and SDP’s. We use these methods to study DP’s in general, and analogous situations for inclusions of value sets.

For example, if (f,g) is an SDP then f(T) − g(S) in F_q[T,S] is known to be reducible. This has interesting consequences. We extend this to DP’s (that are not pairs of exceptional polynomials) and use reducibility to study the relationship between DP’s and SDP’s when f is indecomposable. Additionally, we show that DP’s satisfy (deg f, q^t − 1) = (deg g, q^t − 1) for all suficiently large t with V_f(F_q^t) = V_g(F_q^t). This extends Lenstra’s theorem (Carlitz–Wan conjecture) concerning exceptional polynomials.

Authors