Notes on the Distribution of Roots Modulo a Prime of a Polynomial III

Abstract

Abstract Let f (x) bea monicpolynomialwith integer coefficients and integers r 1,..., r n with 0 ≤ r 1 ≤··· ≤ r n <p the n roots of f (x) ≡ 0mod p for a prime p. We proposed conjectures on the distribution of the point (r 1 /p,...,r n /p) in the previous papers. One aim of this paper is to revise them for a reducible polynomial f (x), and the other is to show that they imply the one-dimensional equidistribution of r 1 /p,...,r n /p for an irreducible polynomial f (x) by a geometric way.