The classical Hensel's lemma is stated as follows: Let $f(x) \in \mathbb{Z}\_p[x]$ and $a \in \mathbb{Z}\_p$ satisfy 
$$
|f(a)|\_p < | f'(a) |\_p^2. 
$$
Then there is a unique $\alpha \in \mathbb{Z}\_p$ such that $f(\alpha)=0$
and $|\alpha - a|\_p < |f'(a)|\_p$.


I would like to know an analogous statement for $f\_1(x,y) \in \mathbb{Z}\_p[x,y]$ and $f\_2(x,y) \in \mathbb{Z}\_p[x,y]$ such that I have
$(\alpha\_1, \alpha\_2) \in \mathbb{Z}\_p^2$ with $f\_1(\alpha\_1, \alpha\_2) = f\_2(\alpha\_1, \alpha\_2)=0$. 


I have two questions regarding Hensel's lemma.


1) Is there a known statement for Hensel's lemma for situation like this where we consider more than one polynomial? 


2) How does one deduce it for $f\_1$ and $f\_2$ as above from the classical Hensel's lemma?


Any comments, hints, references are greatly appreciated! Thank you very much!


PS Here $\mathbb{Z}\_p$ is the $p$ adic integers and $| \cdot |\_p$ is the $p$ adic norm.



A multidimensional version of Hensel's lemma? (for more than one polynomial)
See the accepted answer at <https://math.stackexchange.com/questions/48419/hensels-lemma-and-implicit-function-theorem> for Hensel's lemma for $n$ polynomials in $n$ variables (more general versions are possible). Or see <https://kconrad.math.uconn.edu/blurbs/gradnumthy/multivarhensel.pdf>.


It is not a consequence of the classical one-variable case for one polynomial, just as linear algebra in arbitrary dimensions is not a special case of one dimension. But with the right concepts and notation (e.g., Jacobians) the proof is similar to the one-dimensional case.


