# Sylvester-Gallai type theorems for quadratic polynomials

## TL;DR

This paper proves Sylvester-Gallai type theorems for quadratic polynomials, showing that certain vanishing conditions imply the polynomials lie in a low-dimensional space, with implications for polynomial identity problems.

## Contribution

It establishes new Sylvester-Gallai theorems for quadratic polynomials and confirms conjectures related to polynomial identity testing.

## Key findings

- The span of quadratic polynomials under certain vanishing conditions has bounded dimension.
- A colored version of the Sylvester-Gallai theorem for quadratic polynomials is proved.
- Classifies how quadratic polynomials can vanish simultaneously, extending classical theorems.

## Abstract

We prove Sylvester-Gallai type theorems for quadratic polynomials. Specifically, we prove that if a finite collection $\mathcal Q$, of irreducible polynomials of degree at most $2$, satisfy that for every two polynomials $Q_1,Q_2\in {\mathcal Q}$ there is a third polynomial $Q_3\in{\mathcal Q}$ so that whenever $Q_1$ and $Q_2$ vanish then also $Q_3$ vanishes, then the linear span of the polynomials in ${\mathcal Q}$ 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 [ECCC 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).