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.2022.43
URN: urn:nbn:de:0030-drops-160515
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16051/
Go to the corresponding LIPIcs Volume Portal

Peleg, Shir ; Shpilka, Amir

Robust Sylvester-Gallai Type Theorem for Quadratic Polynomials

pdf-format:
LIPIcs-SoCG-2022-43.pdf (0.7 MB)

Abstract

In this work we extend the robust version of the Sylvester-Gallai theorem, obtained by Barak, Dvir, Wigderson and Yehudayoff, and by Dvir, Saraf and Wigderson, to the case of quadratic polynomials. Specifically, we prove that if {?} ⊂ ℂ[x₁.…,x_n] is a finite set, |{?}| = m, of irreducible quadratic polynomials that satisfy the following condition
There is δ > 0 such that for every Q ∈ {?} there are at least δ m polynomials P ∈ {?} such that whenever Q and P vanish then so does a third polynomial in {?}⧵{Q,P}.
then dim(span) = Poly(1/δ).
The work of Barak et al. and Dvir et al. studied the case of linear polynomials and proved an upper bound of O(1/δ) on the dimension (in the first work an upper bound of O(1/δ²) was given, which was improved to O(1/δ) in the second work).

BibTeX - Entry

@InProceedings{peleg_et_al:LIPIcs.SoCG.2022.43,
  author =	{Peleg, Shir and Shpilka, Amir},
  title =	{{Robust Sylvester-Gallai Type Theorem for Quadratic Polynomials}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{43:1--43:15},
  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/2022/16051},
  URN =		{urn:nbn:de:0030-drops-160515},
  doi =		{10.4230/LIPIcs.SoCG.2022.43},
  annote =	{Keywords: Sylvester-Gallai theorem, quadratic polynomials, Algebraic computation}
}

Keywords: Sylvester-Gallai theorem, quadratic polynomials, Algebraic computation
Collection: 38th International Symposium on Computational Geometry (SoCG 2022)
Issue Date: 2022
Date of publication: 01.06.2022

DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI