Suppose I have $n$ polynomials in $n$ variables $G_j(x_1, \ldots, x_n)$ with integer coefficients. Let $u_j$ be some fixed $p$-adic integers. Consider the system of congruences $$ G_j(\mathbf{x}) \equiv u_j x_j^{-1} \pmod{p^k} $$ for $1 \leq j \leq n$. Let $N(p,k)$ be the number of all solutions $\mathbf{a} \in ((\mathbb{Z}/p^k \mathbb{Z})^*)^n$ to the above system of congruences. I am interested in upper bounds for $N(p;k)$.

I am wondering is it possible to obtain $N(p;k) \ll 1$, where the implicit constant may depend on the degrees and the coefficients of the polynomials but not on $p$ or $k$. If so what conditions do I need for the polynomials $G_j$?

I would greatly appreciate any comments. Thank you very much.