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Algebraic osculation and factorization of sparse polynomials

Research paper by Martin Weimann

Indexed on: 14 Apr '09Published on: 14 Apr '09Published in: Mathematics - Algebraic Geometry



Abstract

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.