computational complexity

, Volume 24, Issue 2, pp 295–331 | Cite as

Equivalence of Polynomial Identity Testing and Polynomial Factorization

  • Swastik KoppartyEmail author
  • Shubhangi Saraf
  • Amir Shpilka


In this paper, we show that the problem of deterministically factoring multivariate polynomials reduces to the problem of deterministic polynomial identity testing. Specifically, we show that given an arithmetic circuit (either explicitly or via black-box access) that computes a multivariate polynomial f, the task of computing arithmetic circuits for the factors of f can be solved deterministically, given a deterministic algorithm for the polynomial identity testing problem (we require either a white-box or a black-box algorithm, depending on the representation of f).

Together with the easy observation that deterministic factoring implies a deterministic algorithm for polynomial identity testing, this establishes an equivalence between these two central derandomization problems of arithmetic complexity.

Previously, such an equivalence was known only for multilinear circuits (Shpilka & Volkovich, 2010).


Arithmetic circuits polynomial identity testing polynomial factorization 

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© Springer Basel 2015

Authors and Affiliations

  • Swastik Kopparty
    • 1
    Email author
  • Shubhangi Saraf
    • 2
  • Amir Shpilka
    • 3
  1. 1.Department of Mathematics & Department of Computer ScienceRutgers UniversityNew BrunswickUSA
  2. 2.Department of Mathematics & Department of Computer ScienceRutgers UniversityNew BrunswickUSA
  3. 3.Department of Computer ScienceTel Aviv UniversityTel AvivIsrael

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