Document Open Access Logo

Derandomization via Symmetric Polytopes: Poly-Time Factorization of Certain Sparse Polynomials

Authors Pranav Bisht , Nitin Saxena

PDF
Thumbnail PDF

File

LIPIcs.FSTTCS.2022.9.pdf
  • Filesize: 0.8 MB
  • 19 pages

Document Identifiers

Author Details

Pranav Bisht
  • Department of Computer Science & Engineering, IIT Kanpur, India
Nitin Saxena
  • Department of Computer Science & Engineering, IIT Kanpur, India

Funding

  • Saxena, Nitin: DST-SERB (CRG/2020/000045) and N. Rama Rao Chair.

Acknowledgements

Pranav thanks Ilya Volkovich and Vishwas Bhargava for useful discussions. The authors would also like to thank the anonymous reviewers for their useful comments that improved the presentation of results.

Cite As Get BibTex

Pranav Bisht and Nitin Saxena. Derandomization via Symmetric Polytopes: Poly-Time Factorization of Certain Sparse Polynomials. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.FSTTCS.2022.9

Abstract

More than three decades ago, after a series of results, Kaltofen and Trager (J. Symb. Comput. 1990) designed a randomized polynomial time algorithm for factorization of multivariate circuits. Derandomizing this algorithm, even for restricted circuit classes, is an important open problem. In particular, the case of s-sparse polynomials, having individual degree d = O(1), is very well-studied (Shpilka, Volkovich ICALP'10; Volkovich RANDOM'17; Bhargava, Saraf and Volkovich FOCS'18, JACM'20). We give a complete derandomization for this class assuming that the input is a symmetric polynomial over rationals. Generally, we prove an s^poly(d)-sparsity bound for the factors of symmetric polynomials over any field. This characterizes the known worst-case examples of sparsity blow-up for sparse polynomial factoring.
To factor f, we use techniques from convex geometry and exploit symmetry (only) in the Newton polytope of f. We prove a crucial result about convex polytopes, by introducing the concept of "low min-entropy", which might also be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Pseudorandomness and derandomization
  • Mathematics of computing → Combinatoric problems
Keywords
  • Multivariate polynomial factorization
  • derandomization
  • sparse polynomials
  • symmetric polynomials
  • factor-sparsity
  • convex polytopes

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Elwyn R Berlekamp. Factoring polynomials over finite fields. Bell System Technical Journal, 46(8):1853-1859, 1967. Google Scholar
  2. Vishwas Bhargava, Shubhangi Saraf, and Ilya Volkovich. Deterministic factorization of sparse polynomials with bounded individual degree. Journal of the ACM (JACM), 67(2):1-28, 2020. (Preliminary version in FOCS 2018). Google Scholar
  3. Markus Bläser and Gorav Jindal. On the complexity of symmetric polynomials. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  4. David G Cantor and Hans Zassenhaus. A new algorithm for factoring polynomials over finite fields. Mathematics of Computation, pages 587-592, 1981. Google Scholar
  5. Prasad Chaugule, Mrinal Kumar, Nutan Limaye, Chandra Kanta Mohapatra, Adrian She, and Srikanth Srinivasan. Schur polynomials do not have small formulas if the determinant doesn't. In Proceedings of the 35th Computational Complexity Conference, pages 1-27, 2020. Google Scholar
  6. Benny Chor and Ronald L Rivest. A knapsack-type public key cryptosystem based on arithmetic in finite fields. IEEE Transactions on Information Theory, 34(5):901-909, 1988. Google Scholar
  7. Chi-Ning Chou, Mrinal Kumar, and Noam Solomon. Closure results for polynomial factorization. Theory of Computing, 15(1):1-34, 2019. Google Scholar
  8. Anuj Dawar and Gregory Wilsenach. Symmetric arithmetic circuits. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. Google Scholar
  9. Anuj Dawar and Gregory Wilsenach. Lower Bounds for Symmetric Circuits for the Determinant. In Mark Braverman, editor, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022), volume 215 of Leibniz International Proceedings in Informatics (LIPIcs), pages 52:1-52:22, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.52.
  10. Richard A DeMillo and Richard J Lipton. A probabilistic remark on algebraic program testing. Technical report, Georgia Inst of Tech Atlanta School of Information and Computer science, 1977. Google Scholar
  11. Pranjal Dutta, Nitin Saxena, and Amit Sinhababu. Discovering the roots: Uniform closure results for algebraic classes under factoring. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 1152-1165, 2018. Google Scholar
  12. Hervé Fournier, Nutan Limaye, Meena Mahajan, and Srikanth Srinivasan. The shifted partial derivative complexity of elementary symmetric polynomials. In International Symposium on Mathematical Foundations of Computer Science, pages 324-335. Springer, 2015. Google Scholar
  13. V. Guruswami and M. Sudan. Improved decoding of reed-solomon and algebraic-geometric codes. In Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280), pages 28-37, 1998. URL: https://doi.org/10.1109/SFCS.1998.743426.
  14. Pavel Hrubeš and Amir Yehudayoff. Homogeneous formulas and symmetric polynomials. Computational Complexity, 20(3):559-578, 2011. Google Scholar
  15. Valentine Kabanets and Russell Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. computational complexity, 13(1-2):1-46, 2004. Google Scholar
  16. Erich Kaltofen. Uniform closure properties of p-computable functions. In Proceedings of the eighteenth annual ACM symposium on Theory of computing, pages 330-337, 1986. Google Scholar
  17. Erich Kaltofen. Single-factor hensel lifting and its application to the straight-line complexity of certain polynomials. In Proceedings of the nineteenth annual ACM symposium on Theory of computing, pages 443-452, 1987. Google Scholar
  18. Erich Kaltofen. Factorization of polynomials given by straight-line programs. Adv. Comput. Res., 5:375-412, 1989. Google Scholar
  19. Erich Kaltofen. Polynomial factorization: a success story. In Proceedings of the 2003 international symposium on Symbolic and algebraic computation, pages 3-4, 2003. Google Scholar
  20. Erich Kaltofen and Barry M Trager. Computing with polynomials given byblack boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators. Journal of Symbolic Computation, 9(3):301-320, 1990. Google Scholar
  21. Adam R Klivans and Daniel Spielman. Randomness efficient identity testing of multivariate polynomials. In Proceedings of the thirty-third annual ACM symposium on Theory of computing, pages 216-223, 2001. Google Scholar
  22. Swastik Kopparty, Shubhangi Saraf, and Amir Shpilka. Equivalence of polynomial identity testing and deterministic multivariate polynomial factorization. In 2014 IEEE 29th Conference on Computational Complexity (CCC), pages 169-180. IEEE, 2014. Google Scholar
  23. Arjen K Lenstra, Hendrik Willem Lenstra, and László Lovász. Factoring polynomials with rational coefficients. Mathematische annalen, 261(ARTICLE):515-534, 1982. Google Scholar
  24. Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. Superpolynomial lower bounds against low-depth algebraic circuits. In FOCS'21, 2021. Google Scholar
  25. Rafael Oliveira. Factors of low individual degree polynomials. computational complexity, 25(2):507-561, 2016. Google Scholar
  26. Øystein Ore. Über höhere kongruenzen. Norsk Mat. Forenings Skrifter, 1(7):15, 1922. Google Scholar
  27. AM Ostrowski. Über die bedeutung der theorie der konvexen polyeder für die formale algebra. Jahresberichte Deutsche Math. Verein, 20:98-99, 1921. (English translated version republished in ACM SIGSAM Bulletin, 33(1):5, 1999). Google Scholar
  28. Andrzej Schinzel. Polynomials with special regard to reducibility, volume 77. Cambridge University Press, 2000. Google Scholar
  29. Jacob T Schwartz. Fast probabilistic algorithms for verification of polynomial identities. Journal of the ACM (JACM), 27(4):701-717, 1980. Google Scholar
  30. Amir Shpilka. Affine projections of symmetric polynomials. Journal of Computer and System Sciences, 65(4):639-659, 2002. Google Scholar
  31. Amir Shpilka and Ilya Volkovich. On the relation between polynomial identity testing and finding variable disjoint factors. In International Colloquium on Automata, Languages, and Programming, pages 408-419. Springer, 2010. Google Scholar
  32. Amir Shpilka and Amir Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Now Publishers Inc, 2010. Google Scholar
  33. Amit Sinhababu and Thomas Thierauf. Factorization of polynomials given by arithmetic branching programs. In 35th Computational Complexity Conference (CCC 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. Google Scholar
  34. Madhu Sudan. Decoding of reed solomon codes beyond the error-correction bound. Journal of complexity, 13(1):180-193, 1997. Google Scholar
  35. Madhu Sudan. Algebra and computation. lecture notes, 1998. Google Scholar
  36. Ilya Volkovich. Deterministically factoring sparse polynomials into multilinear factors and sums of univariate polynomials. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2015. Google Scholar
  37. Ilya Volkovich. On some computations on sparse polynomials. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017. Google Scholar
  38. Joachim von zur Gathen. Who was who in polynomial factorization: 1. In Proceedings of the 2006 international symposium on Symbolic and algebraic computation, page 2, 2006. Google Scholar
  39. Joachim von zur Gathen and Jürgen Gerhard. Modern computer algebra. Cambridge university press, 2013. Google Scholar
  40. Joachim von zur Gathen and Erich Kaltofen. Factoring sparse multivariate polynomials. Journal of Computer and System Sciences, 31(2):265-287, 1985. Google Scholar
  41. Günter M Ziegler. Lectures on polytopes, volume 152. Springer Science & Business Media, 2012. Google Scholar
  42. Richard Zippel. Probabilistic algorithms for sparse polynomials. In International symposium on symbolic and algebraic manipulation, pages 216-226. Springer, 1979. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact