Robust Sylvester-Gallai Type Theorem for Quadratic Polynomials

Shir Peleg, Amir Shpilka

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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 Q ? C[x1...., xn] is a finite set, |Q| = m, of irreducible quadratic polynomials that satisfy the following condition There is d > 0 such that for every Q ? Q there are at least dm polynomials P ? Q such that whenever Q and P vanish then so does a third polynomial in Q \ (Q, P). then dim(span(Q)) = Poly(1/d). The work of Barak et al. and Dvir et al. studied the case of linear polynomials and proved an upper bound of O(1/d) on the dimension (in the first work an upper bound of O(1/d2) was given, which was improved to O(1/d) in the second work).

Original languageEnglish
Title of host publication38th International Symposium on Computational Geometry, SoCG 2022
EditorsXavier Goaoc, Michael Kerber
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772273
DOIs
StatePublished - 1 Jun 2022
Event38th International Symposium on Computational Geometry, SoCG 2022 - Berlin, Germany
Duration: 7 Jun 202210 Jun 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume224

Conference

Conference38th International Symposium on Computational Geometry, SoCG 2022
Country/TerritoryGermany
CityBerlin
Period7/06/2210/06/22

Keywords

  • Algebraic computation
  • Sylvester-Gallai theorem
  • quadratic polynomials

All Science Journal Classification (ASJC) codes

  • Software

Fingerprint

Dive into the research topics of 'Robust Sylvester-Gallai Type Theorem for Quadratic Polynomials'. Together they form a unique fingerprint.

Cite this