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.
Three problems on the interactions of conormal waves are considered. Two are examples which demonstrate that nonlinear spreading of singularities can occur when the waves are conormal. In one case, two of the waves are tangential, and the other wave is transversal to the first two. The third result is a noninteraction theorem. It is shown that under certain conditions, no nonlinear spreading of the singularities will occur.
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