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Improvements of convex-dense factorization of bivariate polynomials

Martin Weimann

TL;DR

This work develops a convex-dense factorization framework for bivariate polynomials by exploiting the Newton polygon geometry through Ostrowski constraints, achieving a complexity of $\tilde{O}(V\,r_0^{\omega-1})$ under a non-degeneracy hypothesis. It introduces a fast valuated analytic-factorization method based on $v_{\lambda}$-adic valuations and the ring $\mathbb{A}_{\lambda}$, enabling sparsity-aware Hensel lifting and divide-and-conquer on slopes to factor in $\mathbb{K}((x))[y]$ with precision costs near $\tilde{O}(d\sigma)$. The approach includes a recombination phase that uses residues, logarithmic derivatives, and linear-algebra to assemble global factors from analytic factors, with the key innovation of minimizing the lower lattice length $r_0$ via affine transforms to further reduce complexity. Together, these methods deliver softly linear performance in favorable polygonal configurations and remove the need for shifts that erase combinatorial constraints, improving practical factoring for sparse or structured polynomials. The results are broadly applicable across characteristics and provide new fast algorithms for valuated analytic factorization and convex-dense recombination in multivariate factorization tasks.

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 $\tilde{\mathcal{O}}(Vr_0^{ω-1} )$ where $V$ is the volume of the polygon and $r_0$ is its minimal lower lattice length. This improves the complexity $\tilde{\mathcal{O}}(d^{ω+1})$ 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.

Improvements of convex-dense factorization of bivariate polynomials

TL;DR

This work develops a convex-dense factorization framework for bivariate polynomials by exploiting the Newton polygon geometry through Ostrowski constraints, achieving a complexity of under a non-degeneracy hypothesis. It introduces a fast valuated analytic-factorization method based on -adic valuations and the ring , enabling sparsity-aware Hensel lifting and divide-and-conquer on slopes to factor in with precision costs near . The approach includes a recombination phase that uses residues, logarithmic derivatives, and linear-algebra to assemble global factors from analytic factors, with the key innovation of minimizing the lower lattice length via affine transforms to further reduce complexity. Together, these methods deliver softly linear performance in favorable polygonal configurations and remove the need for shifts that erase combinatorial constraints, improving practical factoring for sparse or structured polynomials. The results are broadly applicable across characteristics and provide new fast algorithms for valuated analytic factorization and convex-dense recombination in multivariate factorization tasks.

Abstract

We develop a new algorithm for factoring a bivariate polynomial which takes fully advantage of the geometry of the Newton polygon of . Under a non degeneracy hypothesis, the complexity is where is the volume of the polygon and is its minimal lower lattice length. This improves the complexity of the classical algorithms which consider the total degree of as the main complexity indicator. The integer 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 with respect to a slope valuation, a result which has its own interest.
Paper Structure (23 sections, 42 theorems, 95 equations, 5 figures, 2 algorithms)

This paper contains 23 sections, 42 theorems, 95 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Fast convex-dense factorization
  3. Even faster
  4. Fast valuated analytic factorization.
  5. Main lines of the proof of Theorem \ref{['thm:main']}
  6. Organisation of the paper
  7. Fast $v_{\lambda}$-adic factorization
  8. The ring $\mathbb{A}_{\lambda}$ and its fast arithmetic
  9. Fast multiplication in $\tilde{\mathbb{A}}_{\lambda}$.
  10. Fast division in $\mathbb{A}_{\lambda}$.
  11. Fast Hensel lifting in $\mathbb{A}_{\lambda}$.
  12. Fast $v_{\lambda}$-adic Hensel lemma
  13. Fast $v_{\lambda}$-adic factorization
  14. The $\lambda$-defect of straightness
  15. Comparisons between various valuations
  16. ...and 8 more sections

Key Result

Theorem 1

There exists a deterministic algorithm which given $F\in \mathbb{K}[x,y]$ non degenerated, computes the irreducible factorization of $F$ over $\mathbb{K}$ with plus some univariate factorizations over $\mathbb{K}$ whose degree sum is $r$.

Figures (5)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4: Illustrated proof of Proposition \ref{['prop:DGDH']}.
  • Figure 5: Proof of Lemma \ref{['lem:good slope']}. In dark blue the polygon $N(F)$ and in light blue its bounding parallelogram of slope $\lambda_F$.

Theorems & Definitions (97)

  • Definition 1
  • Theorem 1
  • Example 1
  • Definition 2
  • Corollary 1
  • Example 2
  • Example 3
  • Theorem 2
  • Remark 1
  • Remark 2
  • ...and 87 more