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