A generalized Sylvester-Gallai type theorem for quadratic polynomials

Shir Peleg, Amir Shpilka

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

In this work we prove a version of the Sylvester-Gallai 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 QQi∈K Qi vanishes, then the linear span of the polynomials in Q has dimension O(1). This extends the earlier result [33] 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 [33] that studied the case when one quadratic polynomial is in the radical of two other quadratics.

Original languageEnglish
Title of host publication35th Computational Complexity Conference, CCC 2020
EditorsShubhangi Saraf
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771566
DOIs
StatePublished - 1 Jul 2020
Event35th Computational Complexity Conference, CCC 2020 - Virtual, Online, Germany
Duration: 28 Jul 202031 Jul 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume169
ISSN (Print)1868-8969

Conference

Conference35th Computational Complexity Conference, CCC 2020
Country/TerritoryGermany
CityVirtual, Online
Period28/07/2031/07/20

Keywords

  • Algebraic computation
  • Computational complexity
  • Computational geometry

Fingerprint

Dive into the research topics of 'A generalized Sylvester-Gallai type theorem for quadratic polynomials'. Together they form a unique fingerprint.

Cite this