Polynomial time deterministic identity testingalgorithm for $Σ^{[3]}ΠΣΠ^{[2]}$ circuits via Edelstein-Kelly type theorem for quadratic polynomials

$Σ^{3}ΠΣΠ^{d}$电路的秩界与PIT：基于非线性Edelstein-Kelly定理

Rank Bounds and PIT for $Σ^3 ΠΣΠ^d$ circuits via a non-linear Edelstein-Kelly theorem

本文证明非线性Edelstein-Kelly定理，推广至任意常度数多项式，由此得到$Σ^{3}ΠΣΠ^{d}$电路的常数秩界，并首次给出其确定性多项式时间PIT算法。

Sylvester-Gallai type theorems for quadratic polynomials

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 …