[Paper Review] Algebraic osculation and factorization of sparse polynomials

Martin Weimann|ArXiv.org|Apr 1, 2009

Polynomial and algebraic computation28 references3 citations

TL;DR

This paper introduces a novel algorithm for absolute factorization of sparse bivariate polynomials by leveraging algebraic osculation in toric varieties, using residue-based criteria to detect irreducible factors via the Newton polytope's geometry. The key contribution is a deterministic, geometry-aware Hensel lifting that reduces recombinations compared to classical algorithms like Galligo-Rupprecht.

ABSTRACT

We prove a theorem on algebraic osculation and we apply our result to the Computer Algebra problem of polynomial factorization. We consider X a smooth completion of the complex plane and D an effective divisor supported on the boundary of X. Our main result gives explicit conditions equivalent to that a given Cartier divisor on D extends to X. These osculation criterions are expressed with residues. We derive from this result a toric Hensel lifting which permits to compute the absolute factorization of a bivariate polynomial by taking in account the geometry of its Newton polytope. In particular, we reduce the number of possible recombinations when compared to the Galligo-Rupprecht algorithm.

Motivation & Objective

• To address the inefficiency of generic coordinate changes in existing LR-algorithms for sparse polynomials.
• To reduce the number of possible recombination cases in absolute factorization by exploiting Newton polytope geometry.
• To develop a deterministic algorithm for absolute factorization that avoids probabilistic assumptions and asymptotic approximations.
• To extend the applicability of residue-based extension criteria to singular curves via toric resolutions.
• To provide a theoretical foundation for using boundary intersection data and Cartier divisor extension in factorization.

Proposed method

• Uses a smooth toric compactification $X$ of $\mathbb{C}^2$ to embed the curve $C$ defined by $f$, preserving Newton polytope information.
• Applies a residue pairing $\langle \gamma, \Psi \rangle = 0$ for all $\Psi \in H^0(X, \Omega_X^2(D))$ to determine if a Cartier divisor $\gamma$ on $D = \partial X$ extends to $X$.
• Employs Grothendieck residues and Dolbeault $\bar{\partial}$-resolution to compute the cohomological obstruction to divisor extension.
• Reduces the factorization problem to solving a sparse linear system of size $2\operatorname{Vol}(Q) + \deg(\gamma' \cdot \partial X)$ with $\operatorname{Card}(Q \cap \mathbb{Z}^2)$ unknowns.
• Uses Minkowski sum decompositions $N_f = P + Q$ to associate line bundle decompositions $\mathcal{O}_X(C) \simeq \mathcal{L}_P \otimes \mathcal{L}_Q$.
• Applies a toric Hensel lifting process with higher-order implicit function computation (up to order $na^2$) to ensure deterministic detection of factors.

Experimental results

Research questions

• RQ1 Can algebraic osculation criteria based on residues be used to detect irreducible factors of sparse bivariate polynomials?
• RQ2 How can the geometry of the Newton polytope be used to reduce the number of recombinations in absolute factorization?
• RQ3 What is the precise condition under which a Cartier divisor on the boundary divisor $D$ of a toric surface extends to the whole surface?
• RQ4 How does the proposed method compare in complexity and precision to classical Hensel lifting and the Galligo-Rupprecht algorithm?
• RQ5 Can non-toric singularities on the boundary be exploited to further reduce recombinations in factorization?

Key findings

• A Cartier divisor $\gamma$ on $D = \partial X$ extends to $X$ if and only if $\langle \gamma, \Psi \rangle = 0$ for all $\Psi \in H^0(X, \Omega_X^2(D))$, providing a residue-based extension criterion.
• The algorithm reduces the number of possible recombinations compared to the Galligo-Rupprecht algorithm by using polytope geometry instead of generic coordinates.
• For a bidegree-$(a,b)$ polynomial, the method computes $a$ and $b$ implicit functions up to orders $b$ and $a$, respectively, whereas the GR-algorithm computes $a+b$ functions up to order $a+b$.
• The required precision for implicit functions in the new method can be as high as $na^2$, significantly exceeding the $2na = \deg(f)$ bound in classical Hensel lifting.
• When facet polynomials are square-free, the algorithm becomes a vanishing-sum LR-algorithm with deterministic detection via residue conditions.
• For curves with singularities along the boundary, a non-toric resolution $\widetilde{X}$ allows the use of Theorem 1, and added exceptional divisors further reduce the recombination number.