On the distribution of values and zeros of polynomial systems over arbitrary sets

Abstract

Let G1,…,Gn∈Fp[X1,…,Xm] be n polynomials in m variables over the finite field Fp of p elements. A result of É. Fouvry and N.M. Katz shows that under some natural condition, for any fixed ε and sufficiently large prime p the vectors of fractional parts({G1(x)p},…,{Gn(x)p}),x∈Γ, are uniformly distributed in the unit cube [0,1]n for any cube Γ∈[0,p−1]m with the side length h⩾p1/2(logp)1+ε. Here we use this result to show the above vectors remain uniformly distributed, when x runs through a rather general set. We also obtain new results about the distribution of solutions to system of polynomial congruences.