Algebraic Osculation and Factorization of Sparse Polynomials

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 C and D an effective divisor with support ∂X = X \ C. Our main result gives explicit conditions equivalent to that a given Cartier divisor on the subscheme (|D|,OD) 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.