Let G1,...,Gn ∈ Fp[X1,...,Xm] be n polynomials in m vari- ables 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 re- garding the distribution of the vectors by fractional parts

􏰃􏰅G1(x)􏰆 􏰅Gn(x)􏰆􏰄 n m p , · · · , p ∈ T , x ∈ Fp .

We prove refinements of equidistribution, such as bounds for the ball dis- crepancy and variance.