The congruence $ax_1x_2\cdots x_k + bx_{k+1}x_{k+2}\cdots x_{2k} \equiv c \pmod p$

Abstract:

For prime $p$ and integers $a,b,c$ with $p \nmid ab$, we obtain solutions of the congruence \begin{equation*} ax_1x_2 \cdots x_k + bx_{k+1}x_{k+2}\cdots x_{2k}\equiv c \pmod p \end{equation*} in a cube $\mathcal B$ with edge length $B$. For a cube in general position, we show that if $p \nmid abc$ and $k \ge 5$, then the congruence above has a solution in any cube with edge length $B \gg p^{\frac 14 + \frac 1{2(\sqrt {k} +1.95)}+ \epsilon }.$ Estimates are given for the case $p|c$ as well, and improvements are given for small $k$. For cubes cornered at the origin, $1 \le x_i \le B$ for all $i$, we obtain a solution provided only that $B\gg p^{\frac 3{2k} + O\left (\frac k{\log \log p}\right )}.$ Under the assumption of GRH best possible estimates are given. Boxes with unequal edge lengths are also discussed.