License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2022.42
URN: urn:nbn:de:0030-drops-160505
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16050/
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Garg, Abhibhav ; Oliveira, Rafael ; Sengupta, Akash Kumar

Robust Radical Sylvester-Gallai Theorem for Quadratics

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LIPIcs-SoCG-2022-42.pdf (0.7 MB)

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).

BibTeX - Entry

@InProceedings{garg_et_al:LIPIcs.SoCG.2022.42,
  author =	{Garg, Abhibhav and Oliveira, Rafael and Sengupta, Akash Kumar},
  title =	{{Robust Radical Sylvester-Gallai Theorem for Quadratics}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{42:1--42:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16050},
  URN =		{urn:nbn:de:0030-drops-160505},
  doi =		{10.4230/LIPIcs.SoCG.2022.42},
  annote =	{Keywords: Sylvester-Gallai theorem, arrangements of hypersurfaces, locally correctable codes, algebraic complexity, polynomial identity testing, algebraic geometry, commutative algebra}
}

Keywords: Sylvester-Gallai theorem, arrangements of hypersurfaces, locally correctable codes, algebraic complexity, polynomial identity testing, algebraic geometry, commutative algebra
Collection: 38th International Symposium on Computational Geometry (SoCG 2022)
Issue Date: 2022
Date of publication: 01.06.2022

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