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

Let f (x) be a monic polynomial with integer coefficients and 0 ≤ r₁ ≤ ··· ≤ r_n<p its roots modulo a prime p. We generalize a conjecture on the distribution of roots r_i with additional congruence relations r_i ≡ R_i mod L from the case that f has no non-trivial linear relation among roots to the case that f has a non-trivial linear relation.