On the Distribution of Values and Zeros of Polynomial Systems over Arbitrary Sets

Abstract

Let $G_1,..., G_n \in \Fp[X_1,...,X_m]$ 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 $\epsilon$ and sufficiently large prime $p$ the vectors of fractional parts $$ (\{\frac{G_1(\vec{x})}{p}},...,\{\frac{G_n(\vec{x})}{p}}), \qquad \vec{x} \in Γ, $$ are uniformly distributed in the unit cube $[0,1{]}^n$ for any cube $\Gamma \in [0,p-1{]}^m$ with the side length $h\ge p^{1/2}(\log p{)}^{1+\epsilon }$. Here we use this result to show the above vectors remain uniformly distributed, when $\vec{x}$ runs through a rather general set. We also obtain new results about the distribution of solutions to system of polynomial congruences.