Notes on the Distribution of Roots Modulo Primes of a Polynomial IV

For polynomials f (x),g₁(x),g₂(x)over ℤ, we report several observations about the density of primes p for which f (x) is fully splitting at p and { g1(r)p }<{ g2(r)p } \left\{ {{{{g_1}\left( r \right)} \over p}} \right\} < \left\{ {{{{g_2}\left( r \right)} \over p}} \right\} for some root r of f (x) ≡ 0 mod p.