Factoring bivariate polynomials using adjoints

Martin Weimann (1, 2)

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

We relate factorization of bivariate polynomials to singularities of projective plane curves. We prove that adjoint polynomials of a polynomial F ∈ k[x, y] with coefficients in a field k permit to recombinations of the factors of F (0, y) induced by both the absolute and rational factorizations of F , and so without using Hensel lifting. We show in such a way that a fast computation of adjoint polynomials leads to a fast factorization. Our results establish the relations between the algorithms of Duval-Ragot based on locally constant functions and the algorithms of Lecerf and Chèze-Lecerf based on lifting and recombinations. The proof is based on cohomological sequences and residue theory.

Mots clés: Factorization; adjoint polynomials; curves; singularities; residues; cohomology

Domaines: Mathématiques [math]; Algèbre commutative [math.AC]; Géométrie algébrique [math.AG]

Martin Weimann. Factoring bivariate polynomials using adjoints. Journal of Symbolic Computation, 2013, 58, pp.77-98. ⟨10.1016/j.jsc.2013.05.011⟩. ⟨hal-02137322⟩