  We prove a theorem on algebraic osculation and we apply our result to the
Computer Algebra problem of polynomial factorization. We consider X a smooth
completion of the complex plane and D an effective divisor supported on the
boundary of X. Our main result gives explicit conditions equivalent to that a
given Cartier divisor on D extends to X. These osculation criterions are
expressed with residues. We derive from this result a toric Hensel lifting
which permits to compute the absolute factorization of a bivariate polynomial
by taking in account the geometry of its Newton polytope. In particular, we
reduce the number of possible recombinations when compared to the
Galligo-Rupprecht algorithm.
