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Robust Radical Sylvester-Gallai Theorem for Quadratics

Authors Abhibhav Garg , Rafael Oliveira , Akash Kumar Sengupta

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Author Details

Abhibhav Garg
  • Cheriton School of Computer Science, University of Waterloo, Canada
Rafael Oliveira
  • Cheriton School of Computer Science, University of Waterloo, Canada
Akash Kumar Sengupta
  • Department of Mathematics, Columbia University, New York, NY, USA

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Abhibhav Garg, Rafael Oliveira, and Akash Kumar Sengupta. Robust Radical Sylvester-Gallai Theorem for Quadratics. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.42

Abstract

We prove a robust generalization of a Sylvester-Gallai type theorem for quadratic polynomials. More precisely, given a parameter 0 < δ ≤ 1 and a finite collection ℱ of irreducible and pairwise independent polynomials of degree at most 2, we say that ℱ is a (δ, 2)-radical Sylvester-Gallai configuration if for any polynomial F_i ∈ ℱ, there exist δ(|ℱ|-1) polynomials F_j such that |rad (F_i, F_j) ∩ ℱ| ≥ 3, that is, the radical of F_i, F_j contains a third polynomial in the set. We prove that any (δ, 2)-radical Sylvester-Gallai configuration ℱ must be of low dimension: that is dim span_ℂ{ℱ} = poly(1/δ).

Subject Classification

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

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References

  1. 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, New York, NY, USA, 2011. Association for Computing Machinery. URL: https://doi.org/10.1145/1993636.1993705.
  2. Peter Borwein and William OJ Moser. A survey of sylvester’s problem and its generalizations. Aequationes Mathematicae, 40(1):111-135, 1990. Google Scholar
  3. Chi-Ning Chou, Mrinal Kumar, and Noam Solomon. Closure results for polynomial factorization. Theory of Computing, 15(1):1-34, 2019. Google Scholar
  4. J-L Colliot-Thélene, J-J Sansuc, and P Swinnerton-Dyer. Intersections of two quadrics and châtelet surfaces. i. Journal für die reine und angewandte Mathematik, 373:37-107, 1987. Google Scholar
  5. Leonard Eugene Dickson. The points of inflexion of a plane cubic curve. The Annals of Mathematics, 16(1/4):50-66, 1914. Google Scholar
  6. 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, Dagstuhl, DEU, 2021. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.11.
  7. Zeev Dvir. Incidence theorems and their applications. arXiv preprint, 2012. URL: http://arxiv.org/abs/1208.5073.
  8. 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
  9. 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
  10. Tibor Gallai. Solution of problem 4065. American Mathematical Monthly, 51:169-171, 1944. Google Scholar
  11. 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
  12. 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
  13. William Vallance Douglas Hodge and Daniel Pedoe. Methods of Algebraic Geometry: Volume 2. Cambridge University Press, 1994. Google Scholar
  14. 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. Google Scholar
  15. Leroy Milton Kelly. A resolution of the sylvester-gallai problem of j.-p. serre. Discrete & Computational Geometry, 1(2):101-104, 1986. Google Scholar
  16. Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. Superpolynomial lower bounds against low-depth algebraic circuits. Electron. Colloquium Comput. Complex., page 81, 2021. URL: https://eccc.weizmann.ac.il/report/2021/081.
  17. Eberhard Melchior. Uber vielseite der projektiven ebene. Deutsche Math, 5:461-475, 1940. Google Scholar
  18. Rafael Oliveira and Akash Sengupta. Radical sylvester-gallai theorem for cubics. Manuscript, 2021. Google Scholar
  19. Shir Peleg and Amir Shpilka. A generalized sylvester-gallai type theorem for quadratic polynomials. CoRR, abs/2003.05152, 2020. URL: http://arxiv.org/abs/2003.05152.
  20. 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. CoRR, abs/2006.08263, 2020. URL: http://arxiv.org/abs/2006.08263.
  21. Shir Peleg and Amir Shpilka. Robust sylvester-gallai type theorem for quadratic polynomials. CoRR, abs/2202.04932, 2022. URL: http://arxiv.org/abs/2202.04932.
  22. 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. Google Scholar
  23. Jean-Pierre Serre. Advanced problem 5359. Amer. Math. Monthly, 73(1):89, 1966. Google Scholar
  24. Amir Shpilka. Sylvester-gallai type theorems for quadratic polynomials. Discrete Analysis, page 14492, 2020. Google Scholar
  25. Gaurav Sinha. Reconstruction of real depth-3 circuits with top fan-in 2. In 31st Conference on Computational Complexity (CCC 2016). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  26. James Joseph Sylvester. Mathematical question 11851. Educational Times, 59(98):256, 1893. Google Scholar
  27. Endre Szemerédi and William T. Trotter. Extremal problems in discrete geometry. Combinatorica, 3(3):381-392, 1983. Google Scholar
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