# PIT for depth-$4$ circuits and Sylvester-Gallai conjecture for polynomials

## TL;DR

This paper explores extending Sylvester-Gallai-based techniques to develop deterministic polynomial-time blackbox identity testing algorithms for depth-4 circuits with bounded top fanin, achieving partial success and conditional results.

## Contribution

It generalizes Sylvester-Gallai approach from depth-3 to depth-4 circuits, providing a conditional deterministic PIT algorithm and an unconditional algorithm for a subclass.

## Key findings

- Conditional polynomial-time PIT algorithm for depth-4 circuits with bounded top fanin.
- Unconditional polynomial-time algorithm for a specific subclass of depth-4 circuits.
- Extension of Sylvester-Gallai theorem-based methods to higher circuit depths.

## Abstract

This text is a development of a preprint of Ankit Gupta. We present an approach for devising a deterministic polynomial time blackbox identity testing (PIT) algorithm for depth-$4$ circuits with bounded top fanin. This approach is similar to Kayal-Shubhangi approach for depth-$3$ circuits. Kayal and Shubhangi based their algorithm on Sylvester-Gallai-type theorem about linear polynomials. We show how it is possible to generalize this approach to depth-$4$ circuits. However we failed to implement this plan completely. We succeeded to construct a polynomial time deterministic algorithm for depth-$4$ circuits with bounded top fanin and its correctness requires a hypothesis. Also we present a polynomial-time (unconditional) algorithm for some subclass of depth-$4$ circuits with bounded top fanin.