A generalized Sylvester-Gallai type theorem for quadratic polynomials

Abstract

In this work we prove a version of the Sylvester-Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of Σ [3] ΠΣΠ [2] circuits. Specifically, we prove that if a finite set of irreducible quadratic polynomials Q satisfy that for every two polynomials∈ K and whenever Q 1 and Q 2 vanish then also ∏ i∈K Q i vanishes, then the linear span of the polynomials in Q has dimension O(1). This extends the earlier result [Shp19] that showed a similar conclusion when |K| = 1.An important technical step in our proof is a theorem classifying all the possible cases in which a product of quadratic polynomials can vanish when two other quadratic polynomials vanish. I.e., when the product is in the radical of the ideal generates by the two quadratics. This step extends a result from [Shp19] that studied the case when one quadratic polynomial is in the radical of two other quadratics.

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

“…In an earlier paper [PS20] we proved a non-colored version of the conjecture for the case r = 2 and unbounded c. Theorem 1.5 (Theorem 1.7 of [PS20]). There exists a universal constant Λ such that the following holds.…”

“…Our proof, as well as the proof of [PS20], follow the blueprint of the proof in [Shp19]. The starting point is a theorem classifying the possible cases in which a product of quadratic polynomials belong to the radical ideal generated by two other quadratics.…”

“…The starting point is a theorem classifying the possible cases in which a product of quadratic polynomials belong to the radical ideal generated by two other quadratics. We state here the more general theorem of [PS20], that we will use in our proof.…”

“…Theorem 1.10 (Theorem 1.8 in [PS20]). Let {Q k } k∈K , A, B be homogeneous polynomials of degree 2 such that ∏ k∈K Q k ∈ A, B .…”

“…Similarly, when we say that they satisfy Theorem 1.10(ii) (Theorem 1.10(iii)) we mean that there is a reducible quadratic in their linear span (they belong to a 1 , a 2 for two linear forms a 1 , a 2 ). ♦ Given this classification, the analysis in [Shp19,PS20] is based on which case of the theorem each pair of polynomials satisfy. The main difference is that the analysis in our work is considerably more difficult than the analysis in [Shp19, PS20] as, unlike [Shp19], there is no unique polynomial P 3 in the radical of two other polynomials P 1 , P 2 but rather a product of polynomials is in the radical.…”