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Equivalence of Polynomial Identity Testing and Polynomial Factorization


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).

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  1. M. Agrawal (2005). Proving Lower Bounds Via Pseudo-random Generators. In Proceedings of the 25th Annual Conference on Foundations of Software Technology and Theoretical Computer Science, volume 3821 of Lecture Notes in Computer Science, 92–105. Springer-Verlag.

  2. M. Agrawal, N. Kayal & N. Saxena (2004). PRIMES is in P. Annals of Mathematics 160(2), 781–793.

  3. Berlekamp E. R.: Factoring polynomials over large finite fields. Mathematics of Computation 24(111), 713–735 (1970)

    Article  MathSciNet  Google Scholar 

  4. R. A. DeMillo & R. J. Lipton (1978). A Probabilistic Remark on Algebraic Program Testing. Inf. Process. Lett. 7(4), 193–195.

  5. Z. Dvir, A. Shpilka & A. Yehudayoff (2009). Hardness-randomness tradeoffs for bounded depth arithmetic circuits. SIAM J. Computing 39(4), 1279–1293.

  6. Gao S.: Factoring multivariate polynomials via partial differential equations. Mathematics of computation 72(242), 801–822 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. J. von zur Gathen (2006). Who was who in polynomial factorization. In Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation, B.M. Trager, editor, 2–2. ACM Press.

  8. J. von zur Gathen & J. Gerhard (1999). Modern computer algebra. Cambridge University Press.

  9. A. Gupta, P. Kamath, N. Kayal & R. Saptharishi (2013). Arithmetic Circuits: A Chasm at Depth Three. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science, 578–587.

  10. J. Heintz & C. P. Schnorr (1980). Testing Polynomials which Are Easy to Compute (Extended Abstract). In Proceedings of the 12th Annual ACM Symposium on Theory of Computing, 262–272.

  11. Kabanets V., Impagliazzo R.: Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds. Computational Complexity 13(1-2), 1–46 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. E. Kaltofen (1989). Factorization of polynomials given by straight-line programs. In Randomness in Computation, S. Micali, editor, volume 5 of Advances in Computing Research, 375–412. JAI Press, Greenwich CT.

  13. E. Kaltofen (1990). Polynomial factorization 1982–1986. In Computers in Mathematics, D. V. Chudnovsky & R. D. Jenks, editors. Marcel Dekker, New York.

  14. E. Kaltofen (1992). Polynomial factorization 1987–1991. In Proceedings of LATIN 1992, volume 583 of Lecture Notes in Computer Science, 294–313. Springer-Verlag.

  15. Kaltofen E.: Effective Noether irreducibility forms and applications. J. of Computer and System Sciences 50(2), 274–295 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  16. E. Kaltofen (2003). Polynomial factorization: a success story. In Proceedings of ISSAC’03, 3–4. ACM Press.

  17. E. Kaltofen & B. M. Trager (1990). Computing with Polynomials Given by Black Boxes for Their Evaluations: Greatest Common Divisors, Factorization, Separation of Numerators and Denominators. J. of Symbolic Computation 9(3), 301–320.

  18. N. Kayal (2007). Derandomizing some number-theoretic and algebraic algorithms. Ph.D. thesis, Indian Institute of Technology, Kanpur, India.

  19. A. K. Lenstra, H. W. Lenstra & L. Lovász (1982). Factoring polynomials with rational coefficients. Mathematische Annalen 261(4), 515–534.

  20. Mulmuley K., Vazirani U., Vazirani V.: Matching is as easy as matrix inversion. Combinatorica 7(1), 105–113 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  21. J. T. Schwartz (1980). Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27(4), 701–717.

  22. A. Shpilka & I. Volkovich (2010). On the Relation between Polynomial Identity Testing and Finding Variable Disjoint Factors. In Proceedings ICALP 2010, volume 6198 of Lecture Notes in Computer Science, 408–419. Springer-Verlag.

  23. Shpilka A., Yehudayoff A.: Arithmetic Circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science 5(3-4), 207–388 (2010)

    Article  MathSciNet  Google Scholar 

  24. M. Sudan (1999). Algebra and Computation. Lecture notes.

  25. R. Zippel (1979). Probabilistic algorithms for sparse polynomials. In Proceedings of EUROSAM ’79, volume 72 of Lecture Notes in Computer Science, 216–226. Springer-Verlag.

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Correspondence to Swastik Kopparty.

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Kopparty, S., Saraf, S. & Shpilka, A. Equivalence of Polynomial Identity Testing and Polynomial Factorization. comput. complex. 24, 295–331 (2015).

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  • Arithmetic circuits
  • polynomial identity testing
  • polynomial factorization

Subject classification

  • 65Y04