Document Open Access Logo

Uniform Bounds on Product Sylvester-Gallai Configurations

Authors Abhibhav Garg , Rafael Oliveira , Akash Kumar Sengupta

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.SoCG.2025.52.pdf
  • Filesize: 0.72 MB
  • 15 pages
HTML5 Logo HTML (experimental)
LIPIcs.SoCG.2025.52.html

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Computational geometry
Keywords
  • Sylvester-Gallai theorem
  • arrangements of hypersurfaces
  • algebraic complexity
  • polynomial identity testing
  • algebraic geometry
  • commutative algebra

Metrics

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

Abstract

In this work, we explore a non-linear extension of the classical Sylvester-Gallai configuration. Let 𝕂 be an algebraically closed field of characteristic zero, and let ℱ = {F_1, …, F_m} ⊂ 𝕂[x_1, …, x_N] denote a collection of irreducible homogeneous polynomials of degree at most d, where each F_i is not a scalar multiple of any other F_j for i ≠ j. We define ℱ to be a product Sylvester-Gallai configuration if, for any two distinct polynomials F_i, F_j ∈ ℱ, the following condition is satisfied: ∏_{k≠i, j} F_k ∈ rad (F_i, F_j) .
We prove that product Sylvester-Gallai configurations are inherently low dimensional. Specifically, we show that there exists a function λ : ℕ → ℕ, independent of 𝕂, N, and m, such that any product Sylvester-Gallai configuration must satisfy: dim(span_𝕂(ℱ)) ≤ λ(d).
This result generalizes the main theorems from (Shpilka 2019, Peleg and Shpilka 2020, Oliveira and Sengupta 2023), and gets us one step closer to a full derandomization of the polynomial identity testing problem for the class of depth 4 circuits with bounded top and bottom fan-in.

Cite As Get BibTex

Abhibhav Garg, Rafael Oliveira, and Akash Kumar Sengupta. Uniform Bounds on Product Sylvester-Gallai Configurations. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.52

Author Details

Abhibhav Garg
  • University of Waterloo, Canada
Rafael Oliveira
  • University of Waterloo, Canada
Akash Kumar Sengupta
  • University of Waterloo, Canada

Acknowledgements

The authors would like to thank Amir Shpilka and Shir Peleg for several helpful conversations in the course of this work.

References

  1. M. Agrawal and Manindra. Vinay. Arithmetic circuits: A chasm at depth four. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008. Google Scholar
  2. Manindra Agrawal. Proving lower bounds via pseudo-random generators. In FSTTCS 2005: Foundations of Software Technology and Theoretical Computer Science: 25th International Conference, Hyderabad, India, December 15-18, 2005. Proceedings 25, pages 92-105. Springer, 2005. URL: https://doi.org/10.1007/11590156_6.
  3. Manindra Agrawal, Neeraj Kayal, and Nitin Saxena. Primes is in p. Annals of mathematics, pages 781-793, 2004. Google Scholar
  4. 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. URL: https://doi.org/10.1137/130910725.
  5. Tigran Ananyan and Melvin Hochster. Small subalgebras of polynomial rings and stillman’s conjecture. Journal of the American Mathematical Society, 33(1):291-309, 2020. Google Scholar
  6. Robert Andrews and Michael A. Forbes. Ideals, determinants, and straightening: proving and using lower bounds for polynomial ideals. In 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022, pages 389-402, 2022. URL: https://doi.org/10.1145/3519935.3520025.
  7. Boaz Barak, Zeev Dvir, Amir Yehudayoff, and Avi Wigderson. Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes. In Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing, STOC '11, pages 519-528, 2011. URL: https://doi.org/10.1145/1993636.1993705.
  8. M. Beecken, J. Mittmann, and N. Saxena. Algebraic independence and blackbox identity testing. Information and Computation, 222:2-19, 2013. 38th International Colloquium on Automata, Languages and Programming (ICALP 2011). URL: https://doi.org/10.1016/j.ic.2012.10.004.
  9. Peter Borwein and William OJ Moser. A survey of sylvester’s problem and its generalizations. Aequationes Mathematicae, 40(1):111-135, 1990. Google Scholar
  10. Chi-Ning Chou, Mrinal Kumar, and Noam Solomon. Closure results for polynomial factorization. Theory of Computing, 15(1):1-34, 2019. URL: https://doi.org/10.4086/TOC.2019.V015A013.
  11. Pranjal Dutta, Prateek Dwivedi, and Nitin Saxena. Deterministic identity testing paradigms for bounded top-fanin depth-4 circuits. In Proceedings of the 36th Computational Complexity Conference, CCC '21, 2021. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.11.
  12. Zeev Dvir. Incidence theorems and their applications. arXiv preprint arXiv:1208.5073, 2012. URL: https://arxiv.org/abs/1208.5073.
  13. Zeev Dvir, Shubhangi Saraf, and Avi Wigderson. Improved rank bounds for design matrices and a new proof of kelly’s theorem. In Forum of Mathematics, Sigma, volume 2. Cambridge University Press, 2014. Google Scholar
  14. 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. URL: https://doi.org/10.1137/05063605X.
  15. Michael Edelstein and Leroy M Kelly. Bisecants of finite collections of sets in linear spaces. Canadian Journal of Mathematics, 18:375-380, 1966. Google Scholar
  16. N. Elkies, L. Pretorius, and K. Swanepoel. Sylvester-gallai theorems for complex numbers and quaternions. Discrete Comput Geom, 35:361-373, 2006. URL: https://doi.org/10.1007/S00454-005-1226-7.
  17. Paul Erdos, Richard Bellman, Hubert S Wall, James Singer, and Victor Thébault. Problems for solution: 4065-4069. The American Mathematical Monthly, 50(1):65-66, 1943. Google Scholar
  18. Stephen Fenner, Rohit Gurjar, and Thomas Thierauf. A deterministic parallel algorithm for bipartite perfect matching. Communications of the ACM, 62(3):109-115, 2019. URL: https://doi.org/10.1145/3306208.
  19. Michael A Forbes and Amir Shpilka. Explicit noether normalization for simultaneous conjugation via polynomial identity testing. In International Workshop on Approximation Algorithms for Combinatorial Optimization, pages 527-542. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-40328-6_37.
  20. Tibor Gallai. Solution of problem 4065. American Mathematical Monthly, 51:169-171, 1944. Google Scholar
  21. Abhibhav Garg, Rafael Oliveira, Shir Peleg, and Akash Kumar Sengupta. Radical Sylvester-Gallai Theorem for Tuples of Quadratics. In 38th Computational Complexity Conference (CCC 2023), pages 20:1-20:30, 2023. URL: https://doi.org/10.4230/LIPIcs.CCC.2023.20.
  22. Abhibhav Garg, Rafael Oliveira, and Akash Kumar Sengupta. Robust Radical Sylvester-Gallai Theorem for Quadratics. In 38th International Symposium on Computational Geometry (SoCG 2022), pages 42:1-42:13, 2022. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.42.
  23. Zeyu Guo. Variety Evasive Subspace Families. In Valentine Kabanets, editor, 36th Computational Complexity Conference (CCC 2021), volume 200 of Leibniz International Proceedings in Informatics (LIPIcs), pages 20:1-20:33, 2021. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.20.
  24. Ankit Gupta. Algebraic geometric techniques for depth-4 pit & sylvester-gallai conjectures for varieties. In Electron. Colloquium Comput. Complex., volume 21, page 130, 2014. Google Scholar
  25. Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. Arithmetic circuits: A chasm at depth 3. SIAM Journal on Computing, 45(3):1064-1079, 2016. URL: https://doi.org/10.1137/140957123.
  26. Rohit Gurjar and Thomas Thierauf. Linear matroid intersection is in quasi-nc. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 821-830, 2017. URL: https://doi.org/10.1145/3055399.3055440.
  27. Sten Hansen. A generalization of a theorem of sylvester on the lines determined by a finite point set. Mathematica Scandinavica, 16(2):175-180, 1965. Google Scholar
  28. Joos Heintz and Claus-Peter Schnorr. Testing polynomials which are easy to compute. In Proceedings of the twelfth annual ACM Symposium on Theory of Computing, pages 262-272, 1980. Google Scholar
  29. Friedrich Hirzebruch. Arrangements of lines and algebraic surfaces. In Arithmetic and geometry, pages 113-140. Springer, 1983. Google Scholar
  30. Valentine Kabanets and Russell Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. computational complexity, 13:1-46, 2004. URL: https://doi.org/10.1007/S00037-004-0182-6.
  31. Zohar S. Karnin and Amir Shpilka. Black box polynomial identity testing of generalized depth-3 arithmetic circuits with bounded top fan-in. In 2008 23rd Annual IEEE Conference on Computational Complexity, pages 280-291, 2008. URL: https://doi.org/10.1109/CCC.2008.15.
  32. Neeraj Kayal and Shubhangi Saraf. Blackbox polynomial identity testing for depth 3 circuits. In 2009 50th Annual IEEE Symposium on Foundations of Computer Science, pages 198-207. IEEE, 2009. URL: https://doi.org/10.1109/FOCS.2009.67.
  33. Swastik Kopparty, Shubhangi Saraf, and Amir Shpilka. Equivalence of polynomial identity testing and polynomial factorization. computational complexity, 24:295-331, 2015. URL: https://doi.org/10.1007/S00037-015-0102-Y.
  34. Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. Superpolynomial lower bounds against low-depth algebraic circuits. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 804-814. IEEE, 2022. Google Scholar
  35. Eberhard Melchior. Uber vielseite der projektiven ebene. Deutsche Math, 5:461-475, 1940. Google Scholar
  36. Ketan Mulmuley. Geometric complexity theory v: Efficient algorithms for noether normalization. Journal of the American Mathematical Society, 30(1):225-309, 2017. Google Scholar
  37. Rafael Oliveira and Akash Sengupta. Radical sylvester-gallai theorem for cubics. FOCS, 2022. Google Scholar
  38. Rafael Oliveira and Akash Kumar Sengupta. Strong algebras and radical sylvester-gallai configurations. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 95-105, 2024. URL: https://doi.org/10.1145/3618260.3649617.
  39. Shir Peleg and Amir Shpilka. A Generalized Sylvester-Gallai Type Theorem for Quadratic Polynomials. In 35th Computational Complexity Conference (CCC 2020), pages 8:1-8:33, 2020. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.8.
  40. 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. In STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 259-271. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451013.
  41. Shir Peleg and Amir Shpilka. Robust sylvester-gallai type theorem for quadratic polynomials. In 38th International Symposium on Computational Geometry, SoCG 2022, June 7-10, 2022, Berlin, Germany, volume 224, pages 43:1-43:15, 2022. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.43.
  42. L. M. Pretorius and K. J. Swanepoel. The sylvester-gallai theorem, colourings and algebra. Discret. Math., 2009. Google Scholar
  43. Nitin Saxena. Progress on polynomial identity testing. Bull. EATCS, 99:49-79, 2009. Google Scholar
  44. Nitin Saxena. Progress on polynomial identity testing-ii. Perspectives in Computational Complexity: The Somenath Biswas Anniversary Volume, pages 131-146, 2014. Google Scholar
  45. Nitin Saxena and Comandur Seshadhri. From sylvester-gallai configurations to rank bounds: Improved blackbox identity test for depth-3 circuits. Journal of the ACM (JACM), 60(5):1-33, 2013. URL: https://doi.org/10.1145/2528403.
  46. Jean-Pierre Serre. Advanced problem 5359. Amer. Math. Monthly, 73(1):89, 1966. Google Scholar
  47. Amir Shpilka. Sylvester-gallai type theorems for quadratic polynomials. Discrete Analysis, page 14492, 2020. Google Scholar
  48. Amir Shpilka and Amir Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends® in Theoretical Computer Science, 5(3-4):207-388, 2010. URL: https://doi.org/10.1561/0400000039.
  49. James Joseph Sylvester. Mathematical question 11851. Educational Times, 59(98):256, 1893. Google Scholar
  50. Sébastien Tavenas. Improved bounds for reduction to depth 4 and depth 3. Information and Computation, 240:2-11, 2015. URL: https://doi.org/10.1016/J.IC.2014.09.004.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact