2026-07-23T08:35:19Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:16051 2024-03-06T10:56:50Z ddc:004 open_access

Robust Sylvester-Gallai Type Theorem for Quadratic Polynomials Peleg, Shir Shpilka, Amir Sylvester-Gallai theorem quadratic polynomials Algebraic computation 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). Schloss Dagstuhl – Leibniz-Zentrum für Informatik Shir Peleg and Amir Shpilka 2022 Is Part Of LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2022.43 urn:nbn:de:0030-drops-160515 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.43 eng https://creativecommons.org/licenses/by/4.0/legalcode