Notes on the Distribution of Roots Modulo a Prime of a Polynomial VI: Case of Polynomials with Local Roots

Abstract

Let f(x) be a monic integral polynomial of degree n and p a prime number for which f(x) is fully decomposable modulo p. Let integers r1,...,rn be the roots of f(x) mod p with 0 ≤ r1 ≤ ··· ≤ rn < p. Inthis series of papers, we have investigated the distribution of the points (r1,. . .,rn). In the present paper, we present several conjectures concerning the distribution of polynomials in the roots ri.