Elements of Large Order in Prime Finite Fields

Abstract

Given f (x, y) ∈ 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) ∈ F p × F 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 * 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+ (p) , where (p) is an arbitrary function tending to 0 when p goes to ∞.

Elements of large order on varieties over prime finite fields

“…Here we extend the result of [8] to points on general algebraic varieties. Although our results and Conjecture 1.1 do not imply each other, our estimates may be considered as yet an indirect confirmation of this conjecture.…”

“…We note that the results of Voloch [25,26] (and thus those of [8]) are motivated by the following general conjecture due to Poonen (but are quantitatively much weaker): Conjecture 1.1. Let A be a semiabelian variety defined over F q and let X be a closed subvariety of A. Denote Z the union of all translates of positive-dimensional semiabelian varieties over the algebraic closure F q of F q contained in X .…”

“…More recently, it has been shown in [8] that if the zero set of a polynomial f (X, Y ) ∈ Z[X, Y ] has no common components with those of X r − Y s and X r Y s − 1 for any r, s ∈ Z, r, s ≥ 0, then for any function ε(z) with lim z→∞ ε(z) = 0, there is a set of primes p of relative density 1 such that for all but at most C(f ) solutions of the equation…”

“…Another possible relaxation of the original problem is to construct elements x in a given field F q or in its extension of large order ord x, see [1,6,7,8,9,15,19,20,25,26] and references therein. We recall that for a non-zero element x ∈ F q in the algebraic closure F q of F q the order ord x is the smallest positive integer t with x t = 1.…”

“…Although our results and Conjecture 1.1 do not imply each other, our estimates may be considered as yet an indirect confirmation of this conjecture. We expand the method of [7,8] by some new ideas including the use of Hilbert's Nullstellensatz.…”