Improvements of convex-dense factorization of bivariate polynomials

TL;DR

This paper introduces a new algorithm for factoring bivariate polynomials that leverages the Newton polygon's geometry, achieving better complexity bounds than classical methods under certain conditions.

Contribution

The paper presents a novel factorization algorithm that exploits Newton polygon geometry, improving complexity bounds for bivariate polynomial factorization.

Findings

Complexity is $ ilde{O}(Vr_0^{\u03b4-1})$ under non-degeneracy.

Improves classical complexity from $ ilde{O}(d^{7+1})$ to a geometry-based bound.

Provides a new fast factorization method in $7[[x]][y]$ with respect to slope valuation.

Abstract

We develop a new algorithm for factoring a bivariate polynomial $F\in \mathbb{K}[x,y]$ which takes fully advantage of the geometry of the Newton polygon of $F$. Under a non degeneracy hypothesis, the complexity is $\tilde{\mathcal{O}}(Vr_0^{\omega-1} )$ where $V$ is the volume of the polygon and $r_0$ is its minimal lower lattice length. This improves the complexity $\tilde{\mathcal{O}}(d^{\omega+1})$ of the classical algorithms which consider the total degree $d$ of $F$ as the main complexity indicator. The integer $r_0\le d$ reflects some combinatorial constraints imposed by the Newton polygon, giving a reasonable and easy-to-compute upper bound for the number of its indecomposable Minkovski summands of positive volume. The proof is based on a new fast factorization algorithm in $\mathbb{K}[[x]][y]$ with respect to a slope valuation, a result which has its own interest.