Distribution of solutions to systems of congruences in balls

Let G₁,, Gₙ∈ Fₚ[X₁,,Xₘ] be n polynomials in m variables over the finite field Fₚ of p elements. For any sufficiently large prime p and non-trivial bounds for the Weyl sums associated to the non-trivial linear combinations of G=(G₁,, Gₙ), we study various properties regarding the distribution of the vectors by fractional parts {equation*} (\{ G_1(x)/p\},⋯,\{ G_n(x)/p\})∈ T^n,{10pt} x∈ F_p^m. {equation*} We prove refinements of equidistribution, such as bounds for the ball discrepancy and variance.