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 $Z_p$?