TY - GEN

T1 - Sylvester-Gallai type theorems for quadratic polynomials

AU - Shpilka, Amir

N1 - Publisher Copyright:
© 2019 Copyright held by the owner/author(s). Publication rights licensed to ACM.

PY - 2019/6/23

Y1 - 2019/6/23

N2 - 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).

AB - 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).

KW - Arithmetic Circuits

KW - Combinatorics

KW - Polynomial identity testing

UR - http://www.scopus.com/inward/record.url?scp=85068769695&partnerID=8YFLogxK

U2 - 10.1145/3313276.3316341

DO - 10.1145/3313276.3316341

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AN - SCOPUS:85068769695

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 1203

EP - 1214

BT - STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

A2 - Charikar, Moses

A2 - Cohen, Edith

PB - Association for Computing Machinery

Y2 - 23 June 2019 through 26 June 2019

ER -