research-article

Complexity Theory Column 88: Challenges in Polynomial Factorization1

Authors:

Michael A. Forbes,

Amir ShpilkaAuthors Info & Claims

ACM SIGACT News, Volume 46, Issue 4

Pages 32 - 49

https://doi.org/10.1145/2852040.2852051

Published: 01 December 2015 Publication History

Abstract

Algebraic complexity theory studies the complexity of computing (multivariate) polynomials efficiently using algebraic circuits. This succinct representation leads to fundamental algorithmic challenges such as the polynomial identity testing (PIT) problem (decide nonzeroness of the computed polynomial) and the polynomial factorization problem (compute succinct representations of the factors of the circuit). While the Schwartz-Zippel-DeMillo-Lipton Lemma [Sch80,Zip79,DL78] gives an easy randomized algorithm for PIT, randomized algorithms for factorization require more ideas as given by Kaltofen [Kal89]. However, even derandomizing PIT remains a fundamental problem in understanding the power of randomness.

In this column, we survey the factorization problem, discussing the algorithmic ideas as well as the applications to other problems. We then discuss the challenges ahead, in particular focusing on the goal of obtaining deterministic factoring algorithms. While deterministic PIT algorithms have been developed for various restricted circuit classes, there are very few corresponding factoring algorithms. We discuss some recent progress on the divisibility testing problem (test if a given polynomial divides another given polynomial) which captures some of the difficulty of factoring. Along the way we attempt to highlight key challenges whose solutions we hope will drive progress in the area.

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Bhargav CDwivedi PSaxena NMohar BShinkar IO'Donnell R(2024)Learning the Coefficients: A Presentable Version of Border Complexity and Applications to Circuit FactoringProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649743(130-140)Online publication date: 10-Jun-2024
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Chakrabarti SDwivedi ASaxena N(2024)Solving polynomial systems over non-fields and applications to modular polynomial factoringJournal of Symbolic Computation10.1016/j.jsc.2024.102314125(102314)Online publication date: Nov-2024
https://doi.org/10.1016/j.jsc.2024.102314
Zeron EDe Loera J(2024)Computing the Degree of Black-Box Polynomials, with ApplicationsHandbook of Visual, Experimental and Computational Mathematics10.1007/978-3-030-93954-0_54-1(1-16)Online publication date: 1-Jun-2024
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Index Terms

Complexity Theory Column 88: Challenges in Polynomial Factorization1
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Algebraic algorithms
2. Theory of computation
  1. Design and analysis of algorithms

Index terms have been assigned to the content through auto-classification.

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Published In

cover image ACM SIGACT News

ACM SIGACT News Volume 46, Issue 4

December 2015

103 pages

ISSN:0163-5700

DOI:10.1145/2852040

Editor:
Stephen Fenner
University of South Carolina

Issue’s Table of Contents

Copyright © 2015 Copyright is held by the owner/author(s).

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Published: 01 December 2015

Published in SIGACT Volume 46, Issue 4

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Cited By

Bhargav CDwivedi PSaxena NMohar BShinkar IO'Donnell R(2024)Learning the Coefficients: A Presentable Version of Border Complexity and Applications to Circuit FactoringProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649743(130-140)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649743
Chakrabarti SDwivedi ASaxena N(2024)Solving polynomial systems over non-fields and applications to modular polynomial factoringJournal of Symbolic Computation10.1016/j.jsc.2024.102314125(102314)Online publication date: Nov-2024
https://doi.org/10.1016/j.jsc.2024.102314
Zeron EDe Loera J(2024)Computing the Degree of Black-Box Polynomials, with ApplicationsHandbook of Visual, Experimental and Computational Mathematics10.1007/978-3-030-93954-0_54-1(1-16)Online publication date: 1-Jun-2024
https://doi.org/10.1007/978-3-030-93954-0_54-1
Dutta PSaxena NSinhababu A(2022)Discovering the Roots: Uniform Closure Results for Algebraic Classes Under FactoringJournal of the ACM10.1145/351035969:3(1-39)Online publication date: 11-Jun-2022
https://dl.acm.org/doi/10.1145/3510359
Dwivedi AMittal RSaxena N(2021)Efficiently factoring polynomials modulo p4Journal of Symbolic Computation10.1016/j.jsc.2020.10.001104(805-823)Online publication date: May-2021
https://doi.org/10.1016/j.jsc.2020.10.001
Bhargava VSaraf SVolkovich I(2020)Deterministic Factorization of Sparse Polynomials with Bounded Individual DegreeJournal of the ACM10.1145/336566767:2(1-28)Online publication date: 4-May-2020
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Dwivedi AMittal RSaxena NDavenport JWang DKauers MBradford R(2019)Efficiently Factoring Polynomials Modulo p4Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation10.1145/3326229.3326233(139-146)Online publication date: 8-Jul-2019
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Roche DKauers MOvchinnikov ASchost É(2018)What Can (and Can't) we Do with Sparse Polynomials?Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3208976.3209027(25-30)Online publication date: 11-Jul-2018
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