ELEMENTS OF LARGE ORDER IN PRIME FINITE FIELDS

Given f(x, y) is an element of Z[x, y] with no common components with x(a) - y(b) and x(a)y(b) - 1, we prove that for p sufficiently large, with C(f) exceptions, the solutions (x, y) is an element of (F) over bar (p) x (F) over bar (p) of f(x, y) = 0 satisfy ord(x) + ord(y) > c(log p/ log log p)(1/2), where c is a constant and ord(r) is the order of r in the multiplicative group (F) over bar (p)*. Moreover, for most p < N, N being a large number, we prove that, with C(f) exceptions, ord(x) + ord(y) > p(1/4+epsilon(p)), where epsilon(p) is an arbitrary function tending to 0 when p goes to infinity.