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.

  • 2
    $\begingroup$ Such an upper bound certainly doesn't hold in general: take all the $u_j$ equal to $1$, and take $G_j(x) = x_1 \cdots x_{j-1} x_{j+1} \cdots x_n$. Then all $n$ congruences are exactly the same, and the number of solutions is $\big( p^{k-1}(p-1) \big)^{n-1}$. $\endgroup$ – Greg Martin Feb 25 '18 at 20:45
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    $\begingroup$ If the system of equations $G_j =u_jx_j^{-1}$ has finitely many solutions in an algebraically closed field of characteristic zero (e.g. $\mathbb{C}$), then it will have at most that many solutions in any $p$-adic field and your bound will follow. $\endgroup$ – Felipe Voloch Feb 25 '18 at 20:55

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