User:Barto/Hensel's Lemma/Multivariate Composite
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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:
- $f(y)\equiv0 \pmod{b^{k+l}}$
- $y\equiv x \pmod{b^k}$