Factoring bivariate polynomials using adjoints

Abstract

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.