Rank Bounds and PIT for $\Sigma^3 \Pi \Sigma \Pi^d$ circuits via a non-linear Edelstein-Kelly theorem

Abhibhav Garg; Rafael Oliveira; Akash Kumar Sengupta

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arXiv · 2504.14729

Rank Bounds and PIT for $\Sigma^3 \Pi \Sigma \Pi^d$ circuits via a non-linear Edelstein-Kelly theorem

Abhibhav Garg·Rafael Oliveira·Akash Kumar Sengupta

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Abstract

We prove a non-linear Edelstein-Kelly theorem for polynomials of constant degree, fully settling a stronger form of Conjecture 30 in Gupta (2014), and generalizing the main result of Peleg and Shpilka (STOC 2021) from quadratic polynomials to polynomials of any constant degree. As a consequence of our result, we obtain constant rank bounds for depth-4 circuits with top fanin 3 and constant bottom fanin (denoted $\Sigma^{3}\Pi\Sigma\Pi^{d}$ circuits) which compute the zero polynomial. This settles a stronger form of Conjecture 1 in Gupta (2014) when $k=3$, for any constant degree bound; additionally this also makes progress on Conjecture 28 in Beecken, Mittmann, and Saxena (Information \& Computation, 2013). Our rank bounds, when combined with Theorem 2 in Beecken, Mittmann, and Saxena (Information \& Computation, 2013) yield the first deterministic, polynomial time PIT algorithm for $\Sigma^{3}\Pi\Sigma\Pi^{d}$ circuits.

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Abhibhav Garg, Rafael Oliveira, Akash Kumar Sengupta. 2025-04-20. Rank Bounds and PIT for $\Sigma^3 \Pi \Sigma \Pi^d$ circuits via a non-linear Edelstein-Kelly theorem. https://arxiv.org/abs/2504.14729

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