in Multidimensional Hensel lifting, @Hurkyl gave a neat sufficient condition for the existence of $p$-adic liftings in the multidimensional case. I have finally gotten around (but please also see p-adic liftings on SAGE for some added difficulties) to checking my specific set of equations manually for this on SAGE, and in fact the Jacobian in my simplest case has rank 1 less than full rank ie 32 instead of 33 (!!) for 48 variables; and if I'm understanding SAGE's output correctly, the RRE form of the linear equation indicates there is no solution to the Jacobian equation (the last congruence mentioned in @Hurkyl's answer) mod $p$ ...
So I need to check necessary conditions as well. For example one situation where the above conditions are sufficient but not necessary would be where there was some redundancy in my list of polynomial equations. Unfortunately running Groebner basis algorithms on this just crashed my computer because of the sheer size of the computations, so I do not know whether indeed there is any such redundancy yet ...
I know the answer for NSC's in the single-variable case but how does one extend that to many dimensions and an unequal number of conditions?
Thanks in advance for any help!