Algebraic Osculation and Application to Factorization of Sparse Polynomials

Martin Weimann (1, 2)

1 UNICAEN - Université de Caen Normandie
2 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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 the complex plane C^2 and D an effective divisor with support the boundary ∂X = X \ C^2. Our main result gives explicit conditions that are necessary and sufficient for a given Cartier divisor on the subscheme (|D|, O_D) to extend to X. These osculation criterions 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.

Domains: Mathematics [math]; Mathematics [math]; Algebraic Geometry [math.AG]; Mathematics [math]; Complex Variables [math.CV]

Martin Weimann. Algebraic Osculation and Application to Factorization of Sparse Polynomials. Foundations of Computational Mathematics, 2012, 12 (2), pp.173-201. ⟨10.1007/s10208-012-9114-z⟩. ⟨hal-02137318⟩