A multidimensional version of Hensel's lemma? (for more than one polynomial)

Question

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.

Answer 1

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.

Answer 2

Complete local rings are Henselian, and for Henselian rings one has http://jmilne.org/math/Books/ECpup1.pdf Theorem 4.2. Does this answer your question?