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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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