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Rank Bounds and Polynomial-Time PIT for Σ^k Π Σ Π² Circuits

Authors Abhibhav Garg , Rafael Oliveira , Akash Kumar Sengupta , Nir Shalmon , Amir Shpilka

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LIPIcs.CCC.2026.17.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Sylvester-Gallai Theorems
  • Polynomial Identity Testing
  • Strong Algebras

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Abstract

A depth-4 algebraic circuit with top fan-in k and bottom fan-in 2 is a circuit Φ of the form Φ = ∑_{i = 1}^k ∏_{j = 1}^{m_i} Q_{ij}, where the polynomials Q_{ij} ∈ 𝕂[x₁, …, x_n] have degree at most 2. The class of all such circuits is denoted by Σ^k Π Σ Π². We say that the circuit Φ is an identity if it formally computes the zero polynomial. An important parameter of Σ^k Π Σ Π² circuits Φ is their (linear) rank, which is defined as the vector space dimension of the polynomials {Q_{ij}}_{i ∈ [k], j ∈ [m_i]}.
We prove that, when the base field 𝕂 is of characteristic zero, the rank of any (simple and minimal) Σ^k Π Σ Π² identity is upper bounded by a function which depends only on the top fan-in k. This result makes progress on [Beecken et al., 2013], being the first work to establish a bound on the rank of such identities that depends only on the top fan-in. Moreover, when combined with [Beecken et al., 2013], our main result yields the first deterministic, polynomial time PIT algorithm for Σ^k Π Σ Π² circuits.
One of the key components of our proof of the rank bounds is the derivation of an approximate Hansen-type result, which is interesting in its own right. This result can be seen as an algebraic and higher-dimensional analogue of the approximate Sylvester-Gallai result of [Ai et al., 2014], and a distinct approximate fractional Sylvester-Gallai result than the one from [Garg et al., 2023]. Additionally, we prove a robust version of it, in the spirit of the generalization of Hansen’s theorem by [Boaz Barak et al., 2013]. 
This paper is an extended abstract of the full version of the paper, which can be found at [Garg et al., 2026].

Cite As Get BibTex

Abhibhav Garg, Rafael Oliveira, Akash Kumar Sengupta, Nir Shalmon, and Amir Shpilka. Rank Bounds and Polynomial-Time PIT for Σ^k Π Σ Π² Circuits. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CCC.2026.17

Author Details

Abhibhav Garg
  • University of Waterloo, Canada
Rafael Oliveira
  • University of Waterloo, Canada
Akash Kumar Sengupta
  • Rutgers University, Piscataway, NJ, USA
Nir Shalmon
  • Tel Aviv University, Israel
Amir Shpilka
  • Tel Aviv University, Israel

Funding

NS and AS research was funded by the European Union (ERC, EACTP, 101142020). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

Acknowledgements

Part of this work took place while RO was supported by the Simons Institute for the Theory of Computing, and conducted when the author was visiting the Institute. The author would like to thank the generosity, hospitality, and excellent research environment provided by the Simons Institute. Part of this work was done while AKS was postdoctoral fellow at the University of Waterloo.

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