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On Factorization of Sparse Polynomials of Bounded Individual Degree

Aminadav Chuyoon, Amir Shpilka

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.

On Factorization of Sparse Polynomials of Bounded Individual Degree

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 -time algorithm that recovers irreducible -sparse polynomials of individual degree at most from blackbox access to their (not necessarily sparse) product. This partially resolves a question of Dutta-Sinhababu-Thierauf (RANDOM 2024). In particular, if the algorithm runs in polynomial time. 3. Deterministic algorithms for factoring a product of -sparse polynomials of individual degree from blackbox access. Over fields of characteristic zero or sufficiently large characteristic the runtime is ; over arbitrary fields it is . This improves Bhargava-Saraf-Volkovich (JACM 2020), which gives time for a single sparse polynomial. For a single sparse input we obtain 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.
Paper Structure (71 sections, 54 theorems, 44 equations)

This paper contains 71 sections, 54 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Our Results
  4. Related Work
  5. Sparsity of Factors of Sparse Polynomials.
  6. Deterministic factorization and reconstruction.
  7. Divisibility and identity testing.
  8. Proof Overview
  9. Sparse Polynomials of Bounded Individual Degree and Their Factors
  10. Arbitrary Fields.
  11. Zero or Large Characteristic.
  12. Finding Multiquadratic Factors of Products of Sparse Polynomials
  13. Step 1 - Normalization and Recursion.
  14. Step 2 - Obtaining the factorization.
  15. Finding Sparse Divisors of Sparse Polynomials
  16. ...and 56 more sections

Key Result

Theorem 1.8

Let $f\in {\mathcal{P}}(n,s,d)$. The number of non-monomial irreducible factors of $f$, counted with multiplicities, is at most $d\log s$. In particular, the number of divisors of an $s$-sparse polynomial $f$ with bounded individual degree $d$, that are not divisible by any monomial, is at most $2^{

Theorems & Definitions (133)

  • Example 1.1: Example of von1985factoring
  • Definition 1.6
  • Remark 1.7
  • Theorem 1.8: Divisor bound for $(n,s,d)$-sparse polynomials
  • Theorem 1.9: Finding all sparse divisors of $(n,s,d)$-sparse polynomials
  • Theorem 1.10: Factoring products of $(n,s,d)$-sparse polynomials
  • Theorem 1.11: Factoring products of factors of sparse polynomials
  • Theorem 1.12
  • Theorem 1.13
  • Theorem 1.14
  • ...and 123 more