Document Open Access Logo

Variety Evasive Subspace Families

Author Zeyu Guo

PDF

File

Thumbnail PDF
PDF
LIPIcs.CCC.2021.20.pdf
  • Filesize: 0.95 MB
  • 33 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • algebraic complexity
  • dimension reduction
  • Noether normalization
  • polynomial identity testing
  • pseudorandomness
  • varieties

Metrics

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

Abstract

We introduce the problem of constructing explicit variety evasive subspace families. Given a family ℱ of subvarieties of a projective or affine space, a collection ℋ of projective or affine k-subspaces is (ℱ,ε)-evasive if for every 𝒱 ∈ ℱ, all but at most ε-fraction of W ∈ ℋ intersect every irreducible component of 𝒱 with (at most) the expected dimension. The problem of constructing such an explicit subspace family generalizes both deterministic black-box polynomial identity testing (PIT) and the problem of constructing explicit (weak) lossless rank condensers. 
Using Chow forms, we construct explicit k-subspace families of polynomial size that are evasive for all varieties of bounded degree in a projective or affine n-space. As one application, we obtain a complete derandomization of Noether’s normalization lemma for varieties of bounded degree in a projective or affine n-space. In another application, we obtain a simple polynomial-time black-box PIT algorithm for depth-4 arithmetic circuits with bounded top fan-in and bottom fan-in that are not in the Sylvester-Gallai configuration, improving and simplifying a result of Gupta (ECCC TR 14-130).
As a complement of our explicit construction, we prove a lower bound for the size of k-subspace families that are evasive for degree-d varieties in a projective n-space. When n-k = n^Ω(1), the lower bound is superpolynomial unless d is bounded. The proof uses a dimension-counting argument on Chow varieties that parametrize projective subvarieties.

Cite As Get BibTex

Zeyu Guo. Variety Evasive Subspace Families. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 20:1-20:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.CCC.2021.20

Author Details

Zeyu Guo
  • Department of Computer Science, University of Haifa, Israel

Acknowledgements

We thank Nitin Saxena, Noga Ron-Zewi, Amit Sinhababu, and Suryajith Chillara for helpful discussions. We also thank the anonymous reviewers of CCC'21 for their careful reading of our manuscript and their insightful suggestions that helped improve this paper.

References

  1. Manindra Agrawal, Rohit Gurjar, Arpita Korwar, and Nitin Saxena. Hitting-sets for ROABP and sum of set-multilinear circuits. SIAM Journal on Computing, 44(3):669-697, 2015. URL: https://doi.org/10.1137/140975103.
  2. Manindra Agrawal, Chandan Saha, Ramprasad Saptharishi, and Nitin Saxena. Jacobian hits circuits: Hitting sets, lower bounds for depth-d occur-k formulas and depth-3 transcendence degree-k circuits. SIAM Journal on Computing, 45(4):1533-1562, 2016. Google Scholar
  3. Manindra Agrawal and V Vinay. Arithmetic circuits: A chasm at depth four. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, pages 67-75, 2008. Google Scholar
  4. Michael F. Atiyah and I. G. MacDonald. Introduction to Commutative Algebra. Addison-Wesley-Longman, 1969. Google Scholar
  5. Pablo Azcue. On the dimension of the Chow varieties. PhD thesis, Harvard University, 1992. Google Scholar
  6. Malte Beecken, Johannes Mittmann, and Nitin Saxena. Algebraic independence and blackbox identity testing. Information and Computation, 222:2-19, 2013. Google Scholar
  7. Markus Bläser and Anurag Pandey. Polynomial identity testing for low degree polynomials with optimal randomness. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), pages 8:1-8:13, 2020. Google Scholar
  8. Andrej Bogdanov. Pseudorandom generators for low degree polynomials. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pages 21-30, 2005. Google Scholar
  9. Juliette Bruce and Daniel Erman. A probabilistic approach to systems of parameters and Noether normalization. Algebra & Number Theory, 13(9):2081-2102, 2019. Google Scholar
  10. Nader Bshouty. Testers and their applications. In Proceedings of the 5th Conference on Innovations in Theoretical Computer Science, pages 327-352, 2014. Google Scholar
  11. Arthur Cayley. On a new analytical representation of curves in space. The Quarterly Journal of Pure and Applied Mathematics, 3:225-236, 1860. Google Scholar
  12. Arkadev Chattopadhyay, Ankit Garg, and Suhail Sherif. Towards stronger counterexamples to the log-approximate-rank conjecture. arXiv preprint, 2020. URL: http://arxiv.org/abs/2009.02717.
  13. Wei-Liang Chow and B.L. van der Waerden. Zur algebraischen Geometrie. IX. Mathematische Annalen, 113(1):692-704, 1937. Google Scholar
  14. Gil Cohen and Amnon Ta-Shma. Pseudorandom generators for low degree polynomials from algebraic geometry codes. In Electronic Colloquium on Computational Complexity (ECCC), volume 20, page 155, 2013. Google Scholar
  15. John Dalbec and Bernd Sturmfels. Introduction to Chow forms. Invariant Methods in Discrete and Computational Geometry, pages 37-58, 1995. Google Scholar
  16. Richard A. Demillo and Richard J. Lipton. A probabilistic remark on algebraic program testing. Information Processing Letters, 7(4):193-195, 1978. Google Scholar
  17. Thomas W Dubé. A combinatorial proof of the effective Nullstellensatz. Journal of Symbolic Computation, 15(3):277-296, 1993. Google Scholar
  18. Zeev Dvir. Extractors for varieties. Computational Complexity, 21(4):515-572, 2012. Google Scholar
  19. Zeev Dvir, Ariel Gabizon, and Avi Wigderson. Extractors and rank extractors for polynomial sources. Computational Complexity, 18(1):1-58, 2009. Google Scholar
  20. Zeev Dvir, János Kollár, and Shachar Lovett. Variety evasive sets. Computational Complexity, 23(4):509-529, 2014. Google Scholar
  21. Zeev Dvir and Shachar Lovett. Subspace evasive sets. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing, pages 351-358, 2012. Google Scholar
  22. Zeev Dvir and Amir Shpilka. Locally decodable codes with two queries and polynomial identity testing for depth 3 circuits. SIAM Journal on Computing, 36(5):1404-1434, 2007. Google Scholar
  23. David Eisenbud. Commutative Algebra: with a View Toward Algebraic Geometry, volume 150. Springer Science & Business Media, 2013. Google Scholar
  24. David Eisenbud and Joe Harris. On varieties of minimal degree. In Proceedings of Symposia in Pure Mathematics, volume 46, pages 3-13, 1987. Google Scholar
  25. David Eisenbud and Joe Harris. The dimension of the Chow variety of curves. Compositio Mathematica, 83(3):291-310, 1992. Google Scholar
  26. Michael A. Forbes. Polynomial identity testing of read-once oblivious algebraic branching programs. PhD thesis, Massachusetts Institute of Technology, 2014. Google Scholar
  27. Michael A. Forbes and Venkatesan Guruswami. Dimension expanders via rank condensers. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), 2015. Google Scholar
  28. Michael A. Forbes, Ramprasad Saptharishi, and Amir Shpilka. Hitting sets for multilinear read-once algebraic branching programs, in any order. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 867-875, 2014. Google Scholar
  29. Michael A. Forbes and Amir Shpilka. On identity testing of tensors, low-rank recovery and compressed sensing. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing, pages 163-172, 2012. Google Scholar
  30. Michael A. Forbes and Amir Shpilka. Explicit Noether normalization for simultaneous conjugation via polynomial identity testing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), pages 527-542, 2013. Google Scholar
  31. Michael A. Forbes and Amir Shpilka. A PSPACE construction of a hitting set for the closure of small algebraic circuits. In Proceedings of the 50th Annual ACM Symposium on Theory of Computing, pages 1180-1192, 2018. Google Scholar
  32. William Fulton. Young Tableaux: With Applications to Representation Theory and Geometry, volume 35. Cambridge University Press, 1997. Google Scholar
  33. Ariel Gabizon and Ran Raz. Deterministic extractors for affine sources over large fields. Combinatorica, 28(4):415-440, 2008. Google Scholar
  34. Israel M. Gelfand, Mikhail M. Kapranov, and Andrei V. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Birkhäuser, 1994. Google Scholar
  35. Marc Giusti, Klemens Hägele, Grégoire Lecerf, Joël Marchand, and Bruno Salvy. The projective Noether Maple package: computing the dimension of a projective variety. Journal of Symbolic Computation, 30(3):291-307, 2000. Google Scholar
  36. Marc Giusti and Joos Heintz. La d etermination des points isol es et de la dimension d'une vari et e alg ebrique peut se faire en temps polynomial. Computational Algebraic Geometry and Commutative Algebra (Cortona, 1991), pages 216-256, 1993. Google Scholar
  37. Zeyu Guo and Noga Ron-Zewi. Efficient list-decoding with constant alphabet and list sizes. arXiv preprint, 2020. To appear in STOC 2021. URL: http://arxiv.org/abs/2011.05884.
  38. Zeyu Guo, Nitin Saxena, and Amit Sinhababu. Algebraic dependencies and PSPACE algorithms in approximative complexity over any field. Theory of Computing, 15(16):1-30, 2019. Google Scholar
  39. Ankit Gupta. Algebraic geometric techniques for depth-4 PIT & Sylvester-Gallai conjectures for varieties. In Electronic Colloquium on Computational Complexity (ECCC), volume 21, page 130, 2014. Google Scholar
  40. Venkatesan Guruswami and Swastik Kopparty. Explicit subspace designs. Combinatorica, 36(2):161-185, 2016. Google Scholar
  41. Venkatesan Guruswami, Nicolas Resch, and Chaoping Xing. Lossless dimension expanders via linearized polynomials and subspace designs. Combinatorica, pages 1-35, 2021. Google Scholar
  42. Venkatesan Guruswami, Carol Wang, and Chaoping Xing. Explicit list-decodable rank-metric and subspace codes via subspace designs. IEEE Transactions on Information Theory, 62(5):2707-2718, 2016. Google Scholar
  43. Venkatesan Guruswami and Chaoping Xing. List decoding Reed-Solomon, Algebraic-Geometric, and Gabidulin subcodes up to the Singleton bound. In Proceedings of the 45th Annual ACM Symposium on Theory of Computing, pages 843-852, 2013. Google Scholar
  44. Venkatesan Guruswami, Chaoping Xing, and Chen Yuan. Subspace designs based on algebraic function fields. Transactions of the American Mathematical Society, 370(12):8757-8775, 2018. Google Scholar
  45. Joe Harris. Algebraic Geometry: A First Course, volume 133. Springer Science & Business Media, 2013. Google Scholar
  46. Joos Heintz. Definability and fast quantifier elimination in algebraically closed fields. Theoretical Computer Science, 24(3):239-277, 1983. Google Scholar
  47. Joos Heintz and Malte Sieveking. Absolute primality of polynomials is decidable in random polynomial time in the number of variables. In International Colloquium on Automata, Languages, and Programming, pages 16-28, 1981. Google Scholar
  48. David Hilbert. Ueber die Theorie der algebraischen Formen. Mathematische Annalen, 36(4):473-534, 1890. Google Scholar
  49. Gabriela Jeronimo, Teresa Krick, Juan Sabia, and Martín Sombra. The computational complexity of the Chow form. Foundations of Computational Mathematics, 4(1):41-117, 2004. Google Scholar
  50. Michael Joswig and Thorsten Theobald. Polyhedral and Algebraic Methods in Computational Geometry. Springer Science & Business Media, 2013. Google Scholar
  51. Zohar S. Karnin and Amir Shpilka. Black box polynomial identity testing of generalized depth-3 arithmetic circuits with bounded top fan-in. Combinatorica, 31(3):333, 2011. Google Scholar
  52. Adam R. Klivans and Daniel Spielman. Randomness efficient identity testing of multivariate polynomials. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pages 216-223, 2001. Google Scholar
  53. János Kollár. Sharp effective Nullstellensatz. Journal of the American Mathematical Society, pages 963-975, 1988. Google Scholar
  54. János Kollár. Rational Curves on Algebraic Varieties, volume 32. Springer Science & Business Media, 2013. Google Scholar
  55. Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf, and Mary Wootters. Improved decoding of folded Reed-Solomon and multiplicity codes. In Proceedings of the 59th Annual IEEE Symposium on Foundations of Computer Science, pages 212-223, 2018. Google Scholar
  56. Teresa Krick. Straight-line programs in polynomial equation solving. Foundations of Computational Mathematics, 312:96-136, 2002. Google Scholar
  57. Joseph M. Landsberg. Tensors: geometry and applications. Representation theory, 381(402):3, 2012. Google Scholar
  58. Joseph M. Landsberg. Geometric complexity theory: an introduction for geometers. Annali dell'universita'di Ferrara, 61(1):65-117, 2015. Google Scholar
  59. Brian Lehmann. Asymptotic behavior of the dimension of the Chow variety. Advances in Mathematics, 308:815-835, 2017. Google Scholar
  60. Chi-Jen Lu. Hitting set generators for sparse polynomials over any finite fields. In Proceedings of the 27th Conference on Computational Complexity, pages 280-286, 2012. Google Scholar
  61. Partha Mukhopadhyay. Depth-4 identity testing and Noether’s normalization lemma. In Proceedings of the 11th International Computer Science Symposium in Russia, pages 309-323, 2016. Google Scholar
  62. Ketan Mulmuley. Geometric complexity theory V: Efficient algorithms for Noether normalization. Journal of the American Mathematical Society, 30(1):225-309, 2017. Google Scholar
  63. Masayoshi Nagata. Local rings. Interscience Tracts in Pure and Applied Mathematics, 1962. Google Scholar
  64. Emmy Noether. Der Endlichkeitssatz der Invarianten endlicher linearer Gruppen der Charakteristik p. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1926:28-35, 1926. Google Scholar
  65. Shir Peleg and Amir Shpilka. A generalized Sylvester-Gallai type theorem for quadratic polynomials. In Proceedings of the 35th Computational Complexity Conference, 2020. Google Scholar
  66. Shir Peleg and Amir Shpilka. Polynomial time deterministic identity testing algorithm for Σ^[3]ΠΣΠ^[2] circuits via Edelstein-Kelly type theorem for quadratic polynomials. arXiv preprint, 2020. URL: http://arxiv.org/abs/2006.08263.
  67. Nitin Saxena. Progress on polynomial identity testing. Bulletin of the EATCS, 99:49-79, 2009. Google Scholar
  68. Nitin Saxena. Progress on polynomial identity testing - II. Electronic Colloquium on Computational Complexity (ECCC), 20:186, 2013. URL: http://eccc.hpi-web.de/report/2013/186.
  69. Nitin Saxena and Comandur Seshadhri. Blackbox identity testing for bounded top-fanin depth-3 circuits: The field doesn't matter. SIAM Journal on Computing, 41(5):1285-1298, 2012. Google Scholar
  70. Jacob T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. Journal of the ACM, 27(4):701-717, 1980. Google Scholar
  71. Igor R. Shafarevich. Basic Algebraic Geometry 1: Varieties in Projective Space. Springer Science & Business Media, 2013. Google Scholar
  72. Amir Shpilka. Sylvester-Gallai type theorems for quadratic polynomials. In Proceedings of the 51st Annual ACM Symposium on Theory of Computing, pages 1203-1214, 2019. Google Scholar
  73. Amir Shpilka and Amir Yehudayoff. Arithmetic Circuits: A Survey of Recent Results and Open Questions. Now Publishers Inc, 2010. Google Scholar
  74. Richard Zippel. Probabilistic algorithms for sparse polynomials. In International Symposium on Symbolic and Algebraic Manipulation, pages 216-226, 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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact