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A High Dimensional Cramer’s Rule Connecting Homogeneous Multilinear Equations to Hyperdeterminants

Authors Antoine Joux , Anand Kumar Narayanan

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Author Details

Antoine Joux
  • CISPA – Helmholtz Center for Information Security, Saarbrücken, Germany
Anand Kumar Narayanan
  • SandboxAQ, Palo Alto, CA, USA

Funding

  • Joux, Antoine: was supported by the European Union’s H2020 Programme (grant agreement #ERC-669891).

Acknowledgements

We thank the anonymous ITCS 2025 referees for their valuable suggestions.

Cite As Get BibTex

Antoine Joux and Anand Kumar Narayanan. A High Dimensional Cramer’s Rule Connecting Homogeneous Multilinear Equations to Hyperdeterminants. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 62:1-62:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.62

Abstract

We present a new algorithm for solving homogeneous multilinear equations, which are high dimensional generalisations of solving homogeneous linear equations. First, we present a linear time reduction from solving generic homogeneous multilinear equations to computing hyperdeterminants, via a high dimensional Cramer’s rule. Hyperdeterminants are generalisations of determinants, associated with tensors of formats generalising square matrices. Second, we devise arithmetic circuits to compute hyperdeterminants of boundary format tensors. Boundary format tensors are those that generalise square matrices in the strictest sense. Consequently, we obtain arithmetic circuits for solving generic homogeneous boundary format multilinear equations. The complexity as a function of the input dimension varies across boundary format families, ranging from quasi-polynomial to sub exponential. Curiously, the quasi-polynomial complexity arises for families of increasing dimension, including the family of multipartite quantum systems made of d qubits and one qudit. Finally, we identify potential directions to resolve the hardness the hyperdeterminants, notably modulo prime numbers through the cryptographically significant tensor isomorphism complexity class.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • arithmetic circuits
  • tensors
  • determinants
  • hyperdeterminants

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