On Factorization of Sparse Polynomials of Bounded Individual Degree

2026-03-08

AI Summary

This work investigates the factorization of sparse polynomials under individual degree bounds. In the black-box access model, it proposes a deterministic algorithm that efficiently recovers sparse factors and establishes, for the first time, an upper bound on the number of non-monomial irreducible factors of such polynomials. By introducing the notion of “primitive factors,” the approach eliminates dependence on the characteristic of the underlying field, partially resolving an open problem posed by Dutta, Sinhababu, and Thierauf. Combining algebraic structural analysis with black-box reconstruction techniques, the algorithm achieves polynomial or subexponential time complexity over fields of arbitrary characteristic. It significantly improves upon prior results in both univariate and product-input settings—for instance, reducing the complexity from $\mathrm{poly}(n, s^{d^7 \log n})$ to $\mathrm{poly}(n, s^{d^2 \log n})$.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

We study sparse polynomials with bounded individual degree and their factors, obtaining the following structural and algorithmic results. 1. A deterministic polynomial-time algorithm to find all sparse divisors of a sparse polynomial of bounded individual degree, together with the first upper bound on the number of non-monomial irreducible factors of such polynomials. 2. A $\mathrm{poly}(n,s^{d\log \ell})$-time algorithm that recovers $\ell$ irreducible $s$-sparse polynomials of individual degree at most $d$ from blackbox access to their (not necessarily sparse) product. This partially resolves a question of Dutta-Sinhababu-Thierauf (RANDOM 2024). In particular, if $\ell=O(1)$ the algorithm runs in polynomial time. 3. Deterministic algorithms for factoring a product of $s$-sparse polynomials of individual degree $d$ from blackbox access. Over fields of characteristic zero or sufficiently large characteristic the runtime is $\mathrm{poly}(n,s^{d^3\log n})$; over arbitrary fields it is $\mathrm{poly}(n,(d^2)!,s^{d^5\log n})$. This improves Bhargava-Saraf-Volkovich (JACM 2020), which gives $\mathrm{poly}(n,s^{d^7\log n})$ time for a single sparse polynomial. For a single sparse input we obtain $\mathrm{poly}(n,s^{d^2\log n})$ time. 4. Given blackbox access to a product of factors of sparse polynomials of bounded individual degree, we give a deterministic polynomial-time algorithm to find all irreducible sparse multiquadratic factors with multiplicities. This generalizes the algorithms of Volkovich (RANDOM 2015, 2017) and extends the complete-power test of Bisht-Volkovich (CC 2025). To handle arbitrary fields we introduce a notion of primitive divisors that removes characteristic assumptions from most of our algorithms.

Problem

• sparse polynomials
• bounded individual degree
• polynomial factorization
• blackbox access
• irreducible factors

Innovation

• sparse polynomial factorization
• bounded individual degree
• blackbox reconstruction
• deterministic algorithm
• primitive divisors