Solutions to polynomial congruenceswith variables restricted to a box

Abstract

We prove that for any positive integers $k$, $q$, $n$ with $n>N(k)$, integer $c$, and polynomials $f_i(x)$ of degree $k$ whose leading coefficients are relatively prime to $q$, there exists a solution $\underline{x}$ to the congruence $$\sum_{i=1}^n f_i(x_i) \equiv c \pmod q$$ that lies in a cube of side length at least $\max\{q^{1/k},k\}$. Moreover, the result is best possible up to the determination of $N(k)$.