# Necessary and sufficient conditions for Hensel lifting in the multidimensional case

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!

• What is the NSC in the single variable you know ? – user18119 Mar 3 '13 at 22:19
• hi - I meant something like the following: Theorem 5.1 of math.uconn.edu/~kconrad/blurbs/gradnumthy/hensel.pdf although there seems to be a large "folklore" of similar claims ... – GaryMak Mar 4 '13 at 8:48
• by the way this gets pretty close to what I'm looking for ... Bourbaki Comm Alg Ch. III §4.5 Corollary 3 which is slightly mis-cited here: lemma 4.4 of homepages.warwick.ac.uk/~mareg/download/papers/p_adic/… (@Moderators: sorry I'm not sure what I can and cannot post on this site as regards copyright etc so please tell me off if it's unacceptable!!) – GaryMak Mar 4 '13 at 9:22
• Bourbaki says if you have a solution modulo a (explicitely) sufficiently high power of $p$, then you have a solution with coordinates in $\mathbb Z_p$. Kcd's Theorem 5.1 is a necessary condition for the lifting, but I am not sure whether it is usable in you context. – user18119 Mar 4 '13 at 10:15
• thanks - sadly I'm coming to the same conclusion! - I think I will try harder to lift my mod p solutions manually by a couple of levels and hope that the Jacobian behaves ... – GaryMak Mar 4 '13 at 11:08