On Solutions to Some Polynomial Congruences in Small Boxes

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 $p^{1\mathrm{/}2}$, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.