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Efficient Polynomial Identity Testing over Nonassociative Algebras

Authors Partha Mukhopadhyay, C. Ramya , Pratik Shastri

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ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Polynomial identity testing
  • nonassociative algebra
  • arithmetic circuits
  • black-box algorithms
  • white-box algorithms

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Abstract

We design the first efficient polynomial identity testing algorithms over the nonassociative polynomial algebra. In particular, multiplication among the formal variables is commutative but it is not associative. This complements the strong lower bound results obtained over this algebra by Hrubeš, Yehudayoff, and Wigderson [Pavel Hrubes et al., 2010] and Fijalkow, Lagarde, Ohlmann, and Serre [Fijalkow et al., 2021] from the identity testing perspective. Our main results are the following:  
- We construct nonassociative algebras (both commutative and noncommutative) which have no low degree identities. As a result, we obtain the first Amitsur-Levitzki type theorems [A. S. Amitsur and J. Levitzki, 1950] over nonassociative polynomial algebras. As a direct consequence, we obtain randomized polynomial-time black-box PIT algorithms for nonassociative polynomials which allow evaluation over such algebras. 
- On the derandomization side, we give a deterministic polynomial-time identity testing algorithm for nonassociative polynomials given by arithmetic circuits in the white-box setting. Previously, such an algorithm was known with the additional restriction of noncommutativity [Vikraman Arvind et al., 2017]. 
- In the black-box setting, we construct a hitting set of quasipolynomial-size for nonassociative polynomials computed by arithmetic circuits of small depth. Understanding the black-box complexity of identity testing, even in the randomized setting, was open prior to our work.

Cite As Get BibTex

Partha Mukhopadhyay, C. Ramya, and Pratik Shastri. Efficient Polynomial Identity Testing over Nonassociative Algebras. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 56:1-56:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.56

Author Details

Partha Mukhopadhyay
  • Chennai Mathematical Institute, India
C. Ramya
  • The Institute of Mathematical Sciences (a CI of Homi Bhabha National Institute), Chennai, India
Pratik Shastri
  • The Institute of Mathematical Sciences (a CI of Homi Bhabha National Institute), Chennai, India

Funding

  • C. Ramya: Research supported by INSPIRE Faculty Fellowship of DST.

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