On Solutions to Some Polynomial Congruences in Small Boxes

TL;DR

This paper investigates the distribution of solutions to polynomial congruences in small boxes modulo a prime, using bounds on mixed character sums to achieve results in regions smaller than previously possible.

Contribution

It introduces new bounds on solutions to polynomial congruences in small boxes, extending the understanding of solution distribution in regions below the square root of the prime.

Findings

Solutions in boxes with side length below p^{1/2} are nontrivially bounded.

Bounds on mixed character sums are effectively applied to polynomial systems.

Results push the limit of solution distribution analysis in small regions.

Abstract

We use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solution in boxes with the side length below $p^{1/2}$, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.