research-article

Deterministic identity testing paradigms for bounded top-fanin depth-4 circuits

Authors:

Prateek Dwivedi,

Nitin SaxenaAuthors Info & Claims

CCC '21: Proceedings of the 36th Computational Complexity Conference

Article No.: lipics, Page 1

https://doi.org/10.4230/LIPIcs.CCC.2021.11

Published: 29 September 2021 Publication History

Abstract

Polynomial Identity Testing (PIT) is a fundamental computational problem. The famous depth-4 reduction (Agrawal & Vinay, FOCS'08) has made PIT for depth-4 circuits, an enticing pursuit. The largely open special-cases of sum-product-of-sum-of-univariates (Σ^[k]ΠΣ∧) and sum-product-of-constant-degree-polynomials (Σ^[k]ΠΣΠ^δ), for constants k, δ, have been a source of many great ideas in the last two decades. For eg. depth-3 ideas (Dvir & Shpilka, STOC'05; Kayal & Saxena, CCC'06; Saxena & Seshadhri, FOCS'10, STOC'11); depth-4 ideas (Beecken, Mittmann & Saxena, ICALP'11; Saha, Saxena & Saptharishi, Comput.Compl.'13; Forbes, FOCS'15; Kumar & Saraf, CCC'16); geometric Sylvester-Gallai ideas (Kayal & Saraf, FOCS'09; Shpilka, STOC'19; Peleg & Shpilka, CCC'20, STOC'21). We solve two of the basic underlying open problems in this work.

We give the first polynomial-time PIT for (Σ^[k]ΠΣ∧). Further, we give the first quasipolynomial time blackbox PIT for both (Σ^[k]ΠΣ∧) and (Σ^[k]ΠΣΠ^δ). No subexponential time algorithm was known prior to this work (even if k = δ = 3). A key technical ingredient in all the three algorithms is how the logarithmic derivative, and its power-series, modify the top Π-gate to ∧.

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Cited By

Oliveira RSengupta AMohar BShinkar IO'Donnell R(2024)Strong Algebras and Radical Sylvester-Gallai ConfigurationsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649617(95-105)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649617
Garg AOliveira RPeleg SSengupta A(2023)Radical Sylvester-Gallai Theorem for Tuples of QuadraticsProceedings of the conference on Proceedings of the 38th Computational Complexity Conference10.4230/LIPIcs.CCC.2023.20(1-30)Online publication date: 17-Jul-2023
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2023.20
Andrews RForbes MLeonardi SGupta A(2022)Ideals, determinants, and straightening: proving and using lower bounds for polynomial idealsProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520025(389-402)Online publication date: 9-Jun-2022
https://dl.acm.org/doi/10.1145/3519935.3520025

Index Terms

Deterministic identity testing paradigms for bounded top-fanin depth-4 circuits
1. Theory of computation
  1. Computational complexity and cryptography
    1. Algebraic complexity theory

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CCC '21: Proceedings of the 36th Computational Complexity Conference

July 2021

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Oliveira RSengupta AMohar BShinkar IO'Donnell R(2024)Strong Algebras and Radical Sylvester-Gallai ConfigurationsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649617(95-105)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649617
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