Paper Info
Title
Sylvester-Gallai type theorems for quadratic polynomials.
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 [Electronic Colloquium on Computational Complexity (ECCC), 21:130, 2014] 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).
Year
DOI
Venue
2019
10.1145/3313276.3316341
Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing
Keywords
DocType
Volume
Arithmetic Circuits, Combinatorics, polynomial identity testing
Conference
abs/1904.06245
ISSN
ISBN
Citations 
0737-8017
978-1-4503-6705-9
0
PageRank 
References 
Authors
0.34
0
1
Name
Order
Citations
PageRank
Amir Shpilka1109564.27