On the bit-complexity of sparse polynomial and series multiplication
Journal of Symbolic Computation
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We prove a theorem on algebraic osculation and apply our result to the computer algebra problem of polynomial factorization. We consider X a smooth completion of ℂ2 and D an effective divisor with support the boundary ∂X=X∖ℂ2. Our main result gives explicit conditions that are necessary and sufficient for a given Cartier divisor on the subscheme $(|D|,\mathcal{O}_{D})$ to extend to X. These osculation criteria are expressed with residues. When applied to the toric setting, our result gives rise to a new algorithm for factoring sparse bivariate polynomials which takes into account the geometry of the Newton polytope.