Davenport pairs over finite fields:
with Wayne Aitken and Linda Holt
05/05/2003:
Pac J. Math., vol. 215, Sept. 2004.
We call a pair of polynomials f, g in Fq[T]
aDavenport pair (DP) if their value sets (ranges) are equal, for
infinitely many extensions of F Fq. If they are equal
for all extensions of F Fqt (for all t
\ge 1 ), then we say (f, g) is a strong Davenport
pair (SDP). Exceptional polynomials and SDPs are special
cases of DPs. Galois methods have successfully given much
information on exceptional polynomials and SDPs. We use
these to study DPs in general, and analogous situations for
inclusions of value sets. For example, if (f, g) is a DP with
the degree of f prime to the characteristic, then
g=g1og2 with (f, g1) an
SDP with a characterization of their monodromy from projective linear
groups.
For example if (f, g) is a DP then f(T)-g(S) is known to be reducible. This has interesting consequences. We extend this to DPs (that are not pairs of exceptional polynomials) and use reducibility to study the relationship between DPs and SDPs when f is indecomposable. Additionally, we show that DPs satisfy (deg f, q t- 1) = (deg g, q t- 1) for all sufficiently large t with This extends Lenstra's theorem (Carlitz-Wan conjecture) concerning exceptional polynomials.
This paper is a prelude to explicitly understanding the literature on important generating motives over Q and over a finite field. By combining our expertise with Davenport pairs we define relations between motives. Galois stratification shows a suitable generalization of Davenport pairs defines many relations attached to Poincare series. The Denef-Loeser use of Galois stratifications to produce a motivic definition of the Poincare series attached to a question over Q shows that the Galois stratification (Annals '76, and Fried-Jarden, Chaps. 24-26) approach produces a well-defined motive.