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 $N(k)$.
Citation
Todd Cochrane. Konstantinos Kydoniatis. Craig Spencer. "Solutions to polynomial congruenceswith variables restricted to a box." Funct. Approx. Comment. Math. Advance Publication 1 - 9, 2024. https://doi.org/10.7169/facm/2128
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