Sylvester-Gallai type theorems for quadratic polynomials

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Abstract

We prove Sylvester-Gallai type theorems for quadratic polynomials. Specifically, we prove that if a finite collection Q, of irreducible polynomials of degree at most 2, satisfy that for every two polynomials Q1, Q2 ∈ Q there is a third polynomial Q3 ∈ Q so that whenever Q1 and Q2 vanish then also Q3 vanishes, then the linear span of the polynomials in Q has dimension O(1). We also prove a colored version of the theorem: If three finite sets of quadratic polynomials satisfy that for every two polynomials from distinct sets there is a polynomial in the third set satisfying the same vanishing condition then all polynomials are contained in an O(1)-dimensional space. This answers affirmatively two conjectures of Gupta [Gup14] that were raised in the context of solving certain depth-4 polynomial identities. To obtain our main theorems we prove a new result classifying the possible ways that a quadratic polynomial Q can vanish when two other quadratic polynomials vanish. Our proofs also require robust versions of a theorem of Edelstein and Kelly (that extends the Sylvester-Gallai theorem to colored sets).

Original languageEnglish
Title of host publicationSTOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing
EditorsMoses Charikar, Edith Cohen
PublisherAssociation for Computing Machinery
Pages1203-1214
Number of pages12
ISBN (Electronic)9781450367059
DOIs
StatePublished - 23 Jun 2019
Event51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019 - Phoenix, United States
Duration: 23 Jun 201926 Jun 2019

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019
Country/TerritoryUnited States
CityPhoenix
Period23/06/1926/06/19

Keywords

  • Arithmetic Circuits
  • Combinatorics
  • Polynomial identity testing

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