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Rank Bounds and PIT for $Σ^3 ΠΣΠ^d$ circuits via a non-linear Edelstein-Kelly theorem

Abhibhav Garg, Rafael Oliveira, Akash Kumar Sengupta

TL;DR

The paper advances deterministic PIT for depth-4 circuits by proving a non-linear Edelstein-Kelly theorem for constant-degree polynomials, generalizing prior quadratic results to any fixed degree $d$ and thereby bounding the rank of $d$-EK configurations. The authors develop a robust algebraic framework using graded quotients, twisted quotients, and strong Ananyan-Hochster vector spaces to iteratively reduce degree and control irreducibility across quotients, culminating in polynomial-time PIT for $Σ^{3}ΠΣΠ^{d}$ circuits. This yields constant-rank bounds for simple, minimal depth-4 identities and leverages these bounds to derandomize PIT via a reduction to EK-configuration analysis. The work unifies and extends prior approaches (PS20, GOS24) with a streamlined, field-independent method, significantly advancing derandomization prospects for bounded-depth arithmetic circuits. The results have potential implications for broader PIT frontiers, including higher top fan-in regimes, by providing a template for degree-reduction and rank control through EK-configurations.

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 $Σ^{3}ΠΣΠ^{d}$ 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 $Σ^{3}ΠΣΠ^{d}$ circuits.

Rank Bounds and PIT for $Σ^3 ΠΣΠ^d$ circuits via a non-linear Edelstein-Kelly theorem

TL;DR

The paper advances deterministic PIT for depth-4 circuits by proving a non-linear Edelstein-Kelly theorem for constant-degree polynomials, generalizing prior quadratic results to any fixed degree and thereby bounding the rank of -EK configurations. The authors develop a robust algebraic framework using graded quotients, twisted quotients, and strong Ananyan-Hochster vector spaces to iteratively reduce degree and control irreducibility across quotients, culminating in polynomial-time PIT for circuits. This yields constant-rank bounds for simple, minimal depth-4 identities and leverages these bounds to derandomize PIT via a reduction to EK-configuration analysis. The work unifies and extends prior approaches (PS20, GOS24) with a streamlined, field-independent method, significantly advancing derandomization prospects for bounded-depth arithmetic circuits. The results have potential implications for broader PIT frontiers, including higher top fan-in regimes, by providing a template for degree-reduction and rank control through EK-configurations.

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 circuits) which compute the zero polynomial. This settles a stronger form of Conjecture 1 in Gupta (2014) when , 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 circuits.

Paper Structure

This paper contains 41 sections, 37 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. PIT and generalized Edelstein-Kelly configurations
  3. Our Results
  4. Proof overview
  5. Step 1 - Structure theorems:
  6. Step 2 - Finding a set of variables which makes several forms "factor more":
  7. Combinatorial challenges and solutions:
  8. Algebraic challenges and solutions:
  9. Step 3 - Graded quotients and potential function:
  10. Step 4 - Removing assumptions:
  11. Summary of contributions & comparison with previous works
  12. Comparison between our work and PS20b.
  13. Comparison between our work and GOS24.
  14. Related work
  15. Sylvester-Gallai and Edelstein-Kelly configurations
  16. ...and 26 more sections

Key Result

Theorem 1.4

There exists a function $\lambda: {\mathbb N} \rightarrow {\mathbb N}$ such that for any $d$-Edelstein–Kelly configuration $({\mathcal{A}}, {\mathcal{B}}, {\mathcal{C}})$ over a field ${\mathbb K}$ of characteristic $0$, we have

Theorems & Definitions (86)

  • Definition 1.1: Edelstein-Kelly configurations
  • Definition 1.2: Non-linear Edelstein-Kelly configurations
  • Conjecture 1.3: Non-Linear Edelstein-Kelly conjecture
  • Theorem 1.4: Rank bound for EK-configurations
  • Corollary 1.4
  • Corollary 1.4
  • Theorem 1.5
  • Definition 2.1
  • Theorem 2.2: GOS24
  • Proposition 2.3
  • ...and 76 more