Distribution of Roots of Polynomial Congruences

Abstract

For a prime p, we obtain an upper bound on the discrepancy of fractions r/p, where r runs through all of roots modulo p of all monic univariate polynomials of degree d whose vector of coefficients belongs to a d-dimensional box ℬ. The bound is nontrivial starting with boxes ℬ of size |ℬ|≥pd/2+ɛ for any fixed ɛ<0 and sufficiently large p.