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Robust Sylvester-Gallai Type Theorem for Quadratic Polynomials

Authors Shir Peleg , Amir Shpilka

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Author Details

Shir Peleg
  • Tel Aviv University, Israel
Amir Shpilka
  • Tel Aviv University, Israel

Funding

  • Peleg, Shir: The research leading to these results has received funding from the Israel Science Foundation (grant number 514/20) and from the Len Blavatnik and the Blavatnik Family foundation.
  • Shpilka, Amir: The research leading to these results has received funding from the Israel Science Foundation (grant number 514/20) and from the Len Blavatnik and the Blavatnik Family foundation.

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Shir Peleg and Amir Shpilka. Robust Sylvester-Gallai Type Theorem for Quadratic Polynomials. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 43:1-43:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.SoCG.2022.43

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.…,x_n] is a finite set, |{𝒬}| = m, of irreducible quadratic polynomials that satisfy the following condition 
There is δ > 0 such that for every Q ∈ {𝒬} there are at least δ m polynomials P ∈ {𝒬} such that whenever Q and P vanish then so does a third polynomial in {𝒬}⧵{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).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Mathematical analysis
  • Theory of computation → Computational geometry
Keywords
  • Sylvester-Gallai theorem
  • quadratic polynomials
  • Algebraic computation

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