User:Barto/Hensel's Lemma/Multivariate Composite

Theorem
Let $b\neq0,\pm1$ be an integer.

Let $k,N>0$ be positive integers.

Let $f(X) \in \Z[X_1,\ldots,X_N]$ be a polynomial.

Let $x=(x_1,\ldots,x_N) \in \Z^N$ such that:
 * $f(x)\equiv 0 \pmod{b^k}$
 * $\gcd\left( \frac{\partial f}{\partial x_i}(x), b\right )=1$ for some $i\in\{1,\ldots,N\}$

Then for every positive integer $l>0$ there exist, up to a multiple of $b^{k+l}$ exactly $b^{l\cdot(N-1)}$ elements $y\in\Z^N$ such that:
 * $f(y)\equiv0 \pmod{b^{k+l}}$
 * $y\equiv x \pmod{b^k}$