Peleg, Shir ;
Shpilka, Amir
A Generalized SylvesterGallai Type Theorem for Quadratic Polynomials
Abstract
In this work we prove a version of the SylvesterGallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of Σ^{[3]}ΠΣΠ^{[2]} circuits. Specifically, we prove that if a finite set of irreducible quadratic polynomials 𝒬 satisfy that for every two polynomials Q₁,Q₂ ∈ 𝒬 there is a subset 𝒦 ⊂ 𝒬, such that Q₁,Q₂ ∉ 𝒦 and whenever Q₁ and Q₂ vanish then ∏_{Q_i∈𝒦} Q_i vanishes, then the linear span of the polynomials in 𝒬 has dimension O(1). This extends the earlier result [Amir Shpilka, 2019] that showed a similar conclusion when 𝒦 = 1.
An important technical step in our proof is a theorem classifying all the possible cases in which a product of quadratic polynomials can vanish when two other quadratic polynomials vanish. I.e., when the product is in the radical of the ideal generated by the two quadratics. This step extends a result from [Amir Shpilka, 2019] that studied the case when one quadratic polynomial is in the radical of two other quadratics.
BibTeX  Entry
@InProceedings{peleg_et_al:LIPIcs:2020:12560,
author = {Shir Peleg and Amir Shpilka},
title = {{A Generalized SylvesterGallai Type Theorem for Quadratic Polynomials}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {8:18:33},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771566},
ISSN = {18688969},
year = {2020},
volume = {169},
editor = {Shubhangi Saraf},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12560},
URN = {urn:nbn:de:0030drops125606},
doi = {10.4230/LIPIcs.CCC.2020.8},
annote = {Keywords: Algebraic computation, Computational complexity, Computational geometry}
}
17.07.2020
Keywords: 

Algebraic computation, Computational complexity, Computational geometry 
Seminar: 

35th Computational Complexity Conference (CCC 2020)

Issue date: 

2020 
Date of publication: 

17.07.2020 