Equivalence of Polynomial Identity Testing and Polynomial Factorization

Swastik Kopparty*, Shubhangi Saraf, Amir Shpilka

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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).

Original languageEnglish
Article number102
Pages (from-to)295-331
Number of pages37
JournalComputational Complexity
Issue number2
StatePublished - 26 Jun 2015


  • Arithmetic circuits
  • polynomial factorization
  • polynomial identity testing


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