Counting the solutions of $λ_1 x_1^{k_1}+\cdots +λ_t x_t^{k_t}\equiv c\bmod{n}$

Abstract

Given a polynomial $Q(x_1,\cdots, x_t)=λ_1 x_1^{k_1}+\cdots +λ_t x_t^{k_t}$, for every $c\in \mathbb{Z}$ and $n\geq 2$, we study the number of solutions $N_J(Q;c,n)$ of the congruence equation $Q(x_1,\cdots, x_t)\equiv c\bmod{n}$ 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 $a\geq 1$ any integer. As consequences of our main results, we completely solve: the counting problem of $Q(x_i)=\sum\limits_{i\in I}λ_i x_i$ for any prime $p$ and any subset $J$ of $I$; the counting problem of $Q(x_i)=\sum\limits_{i\in I}λ_i x^2_i$ in the case $t=2$ for any $p$ and $J$, and the case $t$ general for any $p$ and $J$ satisfying $\min\{v_p(λ_i)\mid i\in I\}=\min\{v_p(λ_i)\mid i\in J\}$; the counting problem of $Q(x_i)=\sum\limits_{i\in I}λ_i x^k_i$ 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 $\min\{v_p(λ_i)\mid i\in I\}=\min\{v_p(λ_i)\mid i\in J\}$.