Multidimensional Hensel lifting

Question

I have a question about a practical application of (some) generalised form of Hensel's Lemma. I cannot find it stated in an appropriate form in Bourbaki or anywhere else, so here goes ...

Let $p$ be an odd prime: we work over the $p$-adic numbers $Q_p$ with ring of integers $Z_p$ and residue field $F_p$. I have a bunch of quadratic and quartic polynomial equations in N variables, with coefficients in $F_p$, and I have a (non-unique) solution set in $F_p^{N}$.

To simplify things assume that the number of variables N exceeds the total number of equations.

Is it possible to conclude from the non-vanishing of some sort of generalised Jacobian determinant, that my solution set lifts to $Z_p$?

Answer 1

The same proof method works in higher dimension.

Let $F: \mathbb{Z}_p^m \to \mathbb{Z}_p^n$ be our system of polynomial functions.

If $x \in \mathbb{Z}_p^m$ satisfies

$$ F(x) \equiv 0 \pmod {p^e} $$

then

$$ F(x) \equiv p^e y \pmod {p^{e+1}} $$

for some vector $y$. Furthermore, the Taylor expansion about $x$ tells us

$$ F(x + p^e z) \equiv p^e y + p^e dF(x) \cdot z \pmod{p^{e+1}} $$

so as long as

$$ dF(x) \cdot z \equiv -y \pmod p$$

always has a solution for the vector $z$ (e.g. more variables than equations and $dF(x)$ has full rank), then lifts exist.