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

TL;DR

This paper introduces a deterministic polynomial-time algorithm for identity testing of specific algebraic circuits, leveraging a new Edelstein-Kelly type theorem for quadratic polynomials, thus resolving longstanding conjectures.

Contribution

It proves an Edelstein-Kelly type theorem for quadratic polynomials and develops the first deterministic polynomial-time black-box identity test for certain algebraic circuits.

Findings

Proved an Edelstein-Kelly type theorem for quadratic polynomials.

Developed the first deterministic polynomial-time black-box identity testing algorithm for $ ext{Sigma}^{[3]} ext{Pi} ext{Sigma} ext{Pi}^{[2]}$ circuits.

Resolved conjectures by Beecken, Mitmann, Saxena, and Gupta.

Abstract

In this work we resolve conjectures of Beecken, Mitmann and Saxena [BMS13] and Gupta [Gup14], by proving an analog of a theorem of Edelstein and Kelly for quadratic polynomials. As immediate corollary we obtain the first deterministic polynomial time black-box algorithm for testing zeroness of $\Sigma^{[3]}\Pi\Sigma\Pi^{[2]}$ circuits.