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

TL;DR

This paper extends known uniform distribution results of polynomial value vectors over finite fields to more general sets, and provides new insights into the distribution of solutions to polynomial congruences.

Contribution

It generalizes the uniform distribution of polynomial value vectors from cubes to arbitrary sets and introduces new results on the distribution of solutions to polynomial systems.

Findings

Vectors of polynomial values are uniformly distributed over arbitrary sets under certain conditions.

Distribution results hold for sets with side length larger than a specified threshold.

New bounds and distribution properties for solutions to polynomial congruences are established.

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 {\'E}. Fouvry and N. M. Katz shows that under some natural condition, for any fixed $\varepsilon$ and sufficiently large prime $p$ the vectors of fractional parts $$ (\{\frac{G_1(\vec{x})}{p}},...,\{\frac{G_n(\vec{x})}{p}}), \qquad \vec{x} \in \Gamma, $$ are uniformly distributed in the unit cube $[0,1]^n$ for any cube $\Gamma \in [0, p-1]^m$ with the side length $h \ge p^{1/2} (\log p)^{1 + \varepsilon}$. 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.