Davenport pairs over finite fields

Wayne Aitken, Michael D. Fried and Linda M. Holt

Vol. 216 (2004), No. 1, 1–38

DOI: 10.2140/pjm.2004.216.1

Abstract

We call a pair of polynomials f,g ∈ 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) ∈ 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 sufficiently large t with 𝒱_f(𝔽_q^t) = 𝒱_g(𝔽_q^t). This extends Lenstra’s theorem (Carlitz–Wan conjecture) concerning exceptional polynomials.

Milestones

Received: 20 December 2001

Revised: 13 June 2002

Published: 1 September 2004