2026-07-25T06:45:38Z https://drops.dagstuhl.de/oai
oai:drops-oai.dagstuhl.de:23204 2025-10-02T12:42:57Z ddc:004 open_access
Uniform Bounds on Product Sylvester-Gallai Configurations Garg, Abhibhav Oliveira, Rafael Sengupta, Akash Kumar Sylvester-Gallai theorem arrangements of hypersurfaces algebraic complexity polynomial identity testing algebraic geometry commutative algebra In this work, we explore a non-linear extension of the classical Sylvester-Gallai configuration. Let 𝕂 be an algebraically closed field of characteristic zero, and let β„± = {F_1, …, F_m} βŠ‚ 𝕂[x_1, …, x_N] denote a collection of irreducible homogeneous polynomials of degree at most d, where each F_i is not a scalar multiple of any other F_j for i β‰  j. We define β„± to be a product Sylvester-Gallai configuration if, for any two distinct polynomials F_i, F_j ∈ β„±, the following condition is satisfied: ∏_{kβ‰ i, j} F_k ∈ rad (F_i, F_j) . We prove that product Sylvester-Gallai configurations are inherently low dimensional. Specifically, we show that there exists a function Ξ» : β„• β†’ β„•, independent of 𝕂, N, and m, such that any product Sylvester-Gallai configuration must satisfy: dim(span_𝕂(β„±)) ≀ Ξ»(d). This result generalizes the main theorems from (Shpilka 2019, Peleg and Shpilka 2020, Oliveira and Sengupta 2023), and gets us one step closer to a full derandomization of the polynomial identity testing problem for the class of depth 4 circuits with bounded top and bottom fan-in. Schloss Dagstuhl – Leibniz-Zentrum fΓΌr Informatik Abhibhav Garg and Rafael Oliveira and Akash Kumar Sengupta 2025 Is Part Of LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2025.52 urn:nbn:de:0030-drops-232043 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.52 eng https://creativecommons.org/licenses/by/4.0/legalcode