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Uniform Bounds on Product Sylvester-Gallai Configurations

Authors Abhibhav Garg , Rafael Oliveira , Akash Kumar Sengupta

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ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Computational geometry
Keywords
  • Sylvester-Gallai theorem
  • arrangements of hypersurfaces
  • algebraic complexity
  • polynomial identity testing
  • algebraic geometry
  • commutative algebra

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Abstract

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 ℱ = {F_1, …, F_m} ⊂ 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 ≠ j. We define ℱ to be a product Sylvester-Gallai configuration if, for any two distinct polynomials F_i, F_j ∈ ℱ, the following condition is satisfied: ∏_{k≠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 λ : ℕ → ℕ, independent of 𝕂, N, and m, such that any product Sylvester-Gallai configuration must satisfy: dim(span_𝕂(ℱ)) ≤ λ(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.

Cite As Get BibTex

Abhibhav Garg, Rafael Oliveira, and Akash Kumar Sengupta. Uniform Bounds on Product Sylvester-Gallai Configurations. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.52

Author Details

Abhibhav Garg
  • University of Waterloo, Canada
Rafael Oliveira
  • University of Waterloo, Canada
Akash Kumar Sengupta
  • University of Waterloo, Canada

Acknowledgements

The authors would like to thank Amir Shpilka and Shir Peleg for several helpful conversations in the course of this work.

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