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

Let f (x) bea monicpolynomialwith integer coefficients and integers r₁,..., r_n with 0 ≤ r₁ ≤··· ≤ r_n <p the n roots of f (x) ≡ 0mod p for a prime p. We proposed conjectures on the distribution of the point (r₁/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 r₁/p,...,r_n/p for an irreducible polynomial f (x) by a geometric way.