User:Barto/Hensel's Lemma

Draft work, to be placed at Hensel's Lemma some time. Given the number of different versions, we have to think how to structure them: transcluded subpages? Entirely distinct pages?

Hensel's Lemma in $\Z$

First Form

Published at Hensel's Lemma/First Form

Composite Numbers

Published at Hensel's Lemma for Composite Numbers

Singular Point

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 $p^{e+n+1}$.

Multivariate Composite

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:

$f(y)\equiv0 \pmod{b^{k+l}}$

$y\equiv x \pmod{b^k}$