Davenport pairs over finite fields

Abstract. A pair of polynomials f, g ∈ Fq[T] is called a Davenport pair (DP) if their value sets are equal, Vf(Fqt) = Vg(Fqt), for infinitely many extensions of Fq. If they are equal for all extensions of Fq, i.e., for all t ≥ 1, then we say that (f, g) is a strong Davenport pair (SDP). One may consider exceptional polynomials and SDP’s as special cases of DP’s. Exceptional polynomials and SDP’s have been successfully studied using monodromy/Galois-theoretic methods. We use these methods to study DP’s in general, and analogous situations for inclusions of value sets. For example, if (f, g) is a SDP, then f(T) − g(S) ∈ Fq[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 if (f, g) is a DP, then (deg f, qt − 1) = (deg g, qt − 1) for all sufficiently large qt with Vf(Fqt) = Vg(Fqt). This extends Lenstra’s theorem (Carlitz-Wan conjecture) concerning exceptional polynomials. 1