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

Alexey Milovanov

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arXiv · 1902.07201

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

Alexey Milovanov

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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.

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Alexey Milovanov. 2019-02-19. PIT for depth-$4$ circuits and Sylvester-Gallai conjecture for polynomials. https://arxiv.org/abs/1902.07201

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