Equivalence of Polynomial Identity Testing and Polynomial Factorization
- 147 Downloads
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).
KeywordsArithmetic circuits polynomial identity testing polynomial factorization
Unable to display preview. Download preview PDF.
- 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.Google Scholar
- M. Agrawal, N. Kayal & N. Saxena (2004). PRIMES is in P. Annals of Mathematics 160(2), 781–793.Google Scholar
- R. A. DeMillo & R. J. Lipton (1978). A Probabilistic Remark on Algebraic Program Testing. Inf. Process. Lett. 7(4), 193–195.Google Scholar
- Z. Dvir, A. Shpilka & A. Yehudayoff (2009). Hardness-randomness tradeoffs for bounded depth arithmetic circuits. SIAM J. Computing 39(4), 1279–1293.Google Scholar
- 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.Google Scholar
- J. von zur Gathen & J. Gerhard (1999). Modern computer algebra. Cambridge University Press.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- E. Kaltofen (1990). Polynomial factorization 1982–1986. In Computers in Mathematics, D. V. Chudnovsky & R. D. Jenks, editors. Marcel Dekker, New York.Google Scholar
- E. Kaltofen (1992). Polynomial factorization 1987–1991. In Proceedings of LATIN 1992, volume 583 of Lecture Notes in Computer Science, 294–313. Springer-Verlag.Google Scholar
- E. Kaltofen (2003). Polynomial factorization: a success story. In Proceedings of ISSAC’03, 3–4. ACM Press.Google Scholar
- 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.Google Scholar
- N. Kayal (2007). Derandomizing some number-theoretic and algebraic algorithms. Ph.D. thesis, Indian Institute of Technology, Kanpur, India.Google Scholar
- A. K. Lenstra, H. W. Lenstra & L. Lovász (1982). Factoring polynomials with rational coefficients. Mathematische Annalen 261(4), 515–534.Google Scholar
- J. T. Schwartz (1980). Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27(4), 701–717.Google Scholar
- 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.Google Scholar
- M. Sudan (1999). Algebra and Computation. http://people.csail.mit.edu/madhu/FT98/course.html. Lecture notes.
- 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.Google Scholar