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Radical Sylvester-Gallai Theorem for Tuples of Quadratics

Authors Abhibhav Garg , Rafael Oliveira , Shir Peleg, 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
Shir Peleg
  • Blavatnik School of Computer Science, Tel Aviv University, Israel
Akash Kumar Sengupta
  • Department of Mathematics, Columbia University, New York, NY, USA

Acknowledgements

The authors would like to thank Amir Shpilka for several useful discussions throughout the course of this work and for an anonymous reviewer for very helpful comments which helped improve the presentation, as well as to give an alternative proof of Lemma 64, which we give in Appendix A.

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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). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 20:1-20:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.CCC.2023.20

Abstract

We prove a higher codimensional radical Sylvester-Gallai type theorem for quadratic polynomials, simultaneously generalizing [Hansen, 1965; Shpilka, 2020]. Hansen’s theorem is a high-dimensional version of the classical Sylvester-Gallai theorem in which the incidence condition is given by high-dimensional flats instead of lines. We generalize Hansen’s theorem to the setting of quadratic forms in a polynomial ring, where the incidence condition is given by radical membership in a high-codimensional ideal. Our main theorem is also a generalization of the quadratic Sylvester-Gallai Theorem of [Shpilka, 2020]. Our work is the first to prove a radical Sylvester-Gallai type theorem for arbitrary codimension k ≥ 2, whereas previous works [Shpilka, 2020; Shir Peleg and Amir Shpilka, 2020; Shir Peleg and Amir Shpilka, 2021; Garg et al., 2022] considered the case of codimension 2 ideals. Our techniques combine algebraic geometric and combinatorial arguments. A key ingredient is a structural result for ideals generated by a constant number of quadratics, showing that such ideals must be radical whenever the quadratic forms are far apart. Using the wide algebras defined in [Garg et al., 2022], combined with results about integral ring extensions and dimension theory, we develop new techniques for studying such ideals generated by quadratic forms. One advantage of our approach is that it does not need the finer classification theorems for codimension 2 complete intersection of quadratics proved in [Shpilka, 2020; Garg et al., 2022].

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

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References

  1. 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.
  2. Tigran Ananyan and Melvin Hochster. Strength conditions, small subalgebras, and stillman bounds in degree ≤ 4. Transactions of the American Mathematical Society, 373(7):4757-4806, 2020. Google Scholar
  3. M. F. Atiyah and I. G. MacDonald. Introduction to Commutative Algebra. Addison Wesley Publishing Company, 1969. Google Scholar
  4. Boaz Barak, Zeev Dvir, Avi Wigderson, and Amir Yehudayoff. Fractional sylvester-gallai theorems. Proceedings of the National Academy of Sciences, 110, 2013. Google Scholar
  5. 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.
  6. David Bayer and Michael Stillman. On the complexity of computing syzygies. Journal of Symbolic Computation, 6(2):135-147, 1988. URL: https://doi.org/10.1016/S0747-7171(88)80039-7.
  7. Peter Borwein and William OJ Moser. A survey of sylvester’s problem and its generalizations. Aequationes Mathematicae, 40(1):111-135, 1990. Google Scholar
  8. Chi-Ning Chou, Mrinal Kumar, and Noam Solomon. Hardness vs randomness for bounded depth arithmetic circuits. In 33rd Computational Complexity Conference (CCC 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  9. Leonard Eugene Dickson. The points of inflexion of a plane cubic curve. The Annals of Mathematics, 16(1/4):50-66, 1914. Google Scholar
  10. 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.
  11. Zeev Dvir, Ankit Garg, Rafael Oliveira, and József Solymosi. Rank bounds for design matrices with block entries and geometric applications. Discrete Analysis, 2018. Google Scholar
  12. 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
  13. 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
  14. David Eisenbud. Commutative Algebra with a View Toward Algebraic Theory. Springer-Verlag, New York, 1995. Google Scholar
  15. Tibor Gallai. Solution of problem 4065. American Mathematical Monthly, 51:169-171, 1944. Google Scholar
  16. Abhibhav Garg, Rafael Oliveira, and Akash Kumar Sengupta. Robust Radical Sylvester-Gallai Theorem for Quadratics. In 38th International Symposium on Computational Geometry (SoCG 2022), volume 224 of Leibniz International Proceedings in Informatics (LIPIcs), pages 42:1-42:13, 2022. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.42.
  17. Siyao Guo, Tal Malkin, Igor C Oliveira, and Alon Rosen. The power of negations in cryptography. In Theory of Cryptography Conference, pages 36-65. Springer, 2015. Google Scholar
  18. 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.
  19. 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
  20. 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
  21. Friedrich Hirzebruch. Arrangements of lines and algebraic surfaces. In Arithmetic and geometry, pages 113-140. Springer, 1983. Google Scholar
  22. Pavel Hrubeš and Amir Yehudayoff. Monotone separations for constant degree polynomials. Information Processing Letters, 110(1):1-3, 2009. Google Scholar
  23. 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
  24. Leroy Milton Kelly. A resolution of the sylvester-gallai problem of j.-p. serre. Discrete & Computational Geometry, 1(2):101-104, 1986. Google Scholar
  25. 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
  26. Ernst W Mayr and Albert R Meyer. The complexity of the word problems for commutative semigroups and polynomial ideals. Advances in mathematics, 46(3):305-329, 1982. Google Scholar
  27. Eberhard Melchior. Uber vielseite der projektiven ebene. Deutsche Math, 5:461-475, 1940. Google Scholar
  28. Rafael Oliveira and Akash Sengupta. Radical sylvester-gallai theorem for cubics. FOCS, 2022. Google Scholar
  29. 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.
  30. 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.
  31. Shir Peleg and Amir Shpilka. Robust sylvester-gallai type theorem for quadratic polynomials. In 38th International Symposium on Computational Geometry, SoCG 2022, volume 224 of LIPIcs, pages 43:1-43:15, 2022. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.43.
  32. Ran Raz and Avi Wigderson. Monotone circuits for matching require linear depth. Journal of the ACM (JACM), 39(3):736-744, 1992. Google Scholar
  33. Alexander A Razborov. On submodular complexity measures. Boolean Function Complexity,(M. Paterson, Ed.), pages 76-83, 1992. Google Scholar
  34. 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
  35. Jean-Pierre Serre. Advanced problem 5359. Amer. Math. Monthly, 73(1):89, 1966. Google Scholar
  36. Amir Shpilka. Sylvester-gallai type theorems for quadratic polynomials. Discrete Analysis, page 14492, 2020. Google Scholar
  37. 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
  38. The Stacks Project Authors. Stacks Project. Open Source, 2015. URL: http://stacks.math.columbia.edu.
  39. James Joseph Sylvester. Mathematical question 11851. Educational Times, 59(98):256, 1893. Google Scholar
  40. Endre Szemerédi and William T. Trotter. Extremal problems in discrete geometry. Combinatorica, 3(3):381-392, 1983. Google Scholar
  41. Éva Tardos. The gap between monotone and non-monotone circuit complexity is exponential. Combinatorica, 8(1):141-142, 1988. Google Scholar
  42. Leslie G Valiant. Negation can be exponentially powerful. In Proceedings of the eleventh annual ACM symposium on theory of computing, pages 189-196, 1979. Google Scholar
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