I asked a question the other day: Multidimensional Hensel lifting which @Hurkyl kindly and very elegantly answered. A follow-on from this is that I have tried to implement exactly the "algorithm" implicit in his answer in order to solve my problem on SAGE, but without much success. Has anyone out there had any success with $p$-adic lifting problems on SAGE? Unfortunately I need to work over a quadratic extension of $Z_p$ rather than just $Z_p$, which complicates things. The obstacle arises whenever I try to do the division-by-$p^e$ step: the solve() command on SAGE does not seem to deal well with $O(p^n)$ changing from one level of the $p$-adic ring structure to the next.
I have posted a related question on SAGE math forum: http://ask.sagemath.org/question/2309/solving-polynomial-equations-over-p-adic-fields which is more to do with SAGE technicals.
Many thanks for any help!