Counting the solutions of ${\lambda }_1{x}_1^{k_1}+\cdots +{\lambda }_t{x}_t^{k_t}\equiv c\mathrm{mod}n$

Abstract

Given a polynomial $Q(x_1,\cdots ,x_t)={\lambda }_1{x}_1^{k_1}+\cdots +{\lambda }_t{x}_t^{k_t}$, for every $c\in \mathbb{Z}$ and $n\ge 2$, we study the number of solutions $N_J(Q;c,n)$ of the congruence equation $Q(x_1,\cdots ,x_t)\equiv c\mathrm{mod}n$ in $(\mathbb{Z}/n\mathbb{Z}{)}^t$ such that $x_i\in (\mathbb{Z}/n\mathbb{Z}{)}^{\times }$ for $i\in J\subseteq I=\{1,\cdots ,t\}$. We deduce formulas and an algorithm to study $N_J(Q;c,p^a)$ for $p$ any prime number and $a\ge 1$ any integer. As consequences of our main results, we completely solve: the counting problem of $Q(x_i)=\sum _{i\in I}{\lambda }_i{x}_i$ for any prime $p$ and any subset $J$ of $I$; the counting problem of $Q(x_i)=\sum _{i\in I}{\lambda }_i{x}_i^2$ in the case $t=2$ for any $p$ and $J$, and the case $t$ general for any $p$ and $J$ satisfying $\min \{v_p({\lambda }_i)\mid i\in I\}=\min \{v_p({\lambda }_i)\mid i\in J\}$; the counting problem of $Q(x_i)=\sum _{i\in I}{\lambda }_i{x}_i^k$ in the case $t=2$ for any $p\nmid k$ and any $J$, and in the case $t$ general for any $p\nmid k$ and $J$ satisfying $\min \{v_p({\lambda }_i)\mid i\in I\}=\min \{v_p({\lambda }_i)\mid i\in J\}$.

22 pages