User:Barto/Hensel's Lemma/Singular Point

Theorem

Let $p$ be a prime number.

Let $k>0$ be a positive integer.

Let $f(X) \in \Z[X]$ be a polynomial.

Let $x_0\in\Z$ such that:

$f(x_0)\equiv 0 \pmod{p^{2e+1}}$

where $e=\nu_p(f'(x_0))$.

Then for every positive integer $n>0$ there exists an integer $x_{n}$ such that:

$x_{n}\equiv x_{n-1}\pmod{p^{e+n}}$

$f(x_{n})\equiv 0 \pmod{p^{2e+1+n}}$

Moreover, each such $x_{n}$ is unique up to a multiple of $p^{e+n+1}$.