Factoring bivariate polynomials using adjoints

TL;DR

This paper introduces a novel approach to factor bivariate polynomials by leveraging adjoint polynomials and singularity analysis, eliminating the need for Hensel's lifting and connecting existing algorithms through cohomological methods.

Contribution

It demonstrates that adjoint polynomials can efficiently solve factor recombination problems and establishes a theoretical link between two prominent factorization algorithms.

Findings

Adjoint polynomials enable faster polynomial factorization.

The method avoids Hensel's lifting by using singularity-based techniques.

A connection between Duval-Ragot and Chèze-Lecerf algorithms is established.

Abstract

One relates factorization of bivariate polynomials to singularities of projective plane curves. One proves that adjoint polynomials permit to solve the recombinations of the modular factors induced by the absolute and rational factorizations, and so without using Hensel's lifting. One establishes in such a way the relations between the algorithm of Duval-Ragot (locally constant functions) and of Ch\`eze-Lecerf (lifting and recombinations), and one shows that a fast computation of adjoint polynomials leads to a fast factorization. The proof is based on cohomological sequences and residue theory.