On Factorization of Sparse Polynomials of Bounded Individual Degree

TL;DR

This paper develops new deterministic algorithms for factoring sparse polynomials with bounded individual degree, providing structural bounds and efficient methods for identifying factors and divisors over various fields.

Contribution

It introduces the first polynomial-time algorithms for finding all sparse divisors and factors of such polynomials, improving previous bounds and extending applicability to arbitrary fields.

Findings

Deterministic polynomial-time algorithm for all sparse divisors.

Partial resolution of a question on recovering factors from product.

Improved algorithms for factoring products of sparse polynomials.

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.