Explicit formulas for strong Davenport pairs
Antonia Wilson Bluher · Acta Arithmetica · 2004
A pair of separable polynomials (g, h) over a finite field F is said to be a strong Davenport pair if g(K) = h(K) for all finite extension fields K/F , where g(K) = { g(a) | a ∈ K }. We construct examples for which g is a projective or additive polynomial, and we find a factorization of g(x)−h(y). In addition, we prove that |g−1(a)∩K| = |h−1(a) ∩ K| for all a ∈ K. Though the existence of such examples was known to Fried, the explicit formulas for g and h are new, as are our methods of proof. This article also contains new results about additive and projective polynomials. 2000 Mathematics Subject Classification: 12E10, 12E05