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Fixed-Parameter Degree Bounds and Complexity of the Orbit Closure Intersection Problem for Tensors

Authors M. Levent Doğan , John Maar , Rafael Oliveira , Youming Qiao

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LIPIcs.CCC.2026.32.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Computing methodologies → Algebraic algorithms
Keywords
  • computational invariant theory
  • geometric complexity theory
  • orbit closure intersection problem

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Abstract

The orbit closure intersection problem for a reductive group action is a geometric relaxation of the orbit equality problem. These problems capture a range of isomorphism, degeneration and identity-testing problems in computational complexity. We study this problem for the tensor action of G = SL_n(𝕂) x SL_n(𝕂) x SL_m(𝕂) on 𝒱 = 𝕂ⁿ⊗𝕂ⁿ⊗𝕂^m (equivalently, on m-tuples of n× n-matrices) where the base field is algebraically closed with characteristic zero. 
We focus on the fixed-parameter regime where m is constant and n is allowed to grow. The case of m = 3 is already interesting in the context of tensor rank and matrix multiplication. Prior to our work, only a special case of this problem, namely when one of the input tensors is the zero-tensor (this corresponds to the null cone problem for the tensor action), was known to be solvable in polynomial time (Bürgisser-Franks-Garg-Oliveira-Walter-Wigderson, FOCS'19).
Our main result is to show that the orbit closure intersection problem for the above action can be solved in randomized polynomial time. This is achieved by the following new ingredients:  
1) We prove an explicit fixed-parameter bound on the degrees needed to generate the invariant ring for the tensor action: for every fixed m, these bounds are polynomial in n. 
2) We prove that there is a succinct encoding of the invariants: we construct a uniform, polynomial-sized arithmetic circuit generating all invariants up to the required degree.

Cite As Get BibTex

M. Levent Doğan, John Maar, Rafael Oliveira, and Youming Qiao. Fixed-Parameter Degree Bounds and Complexity of the Orbit Closure Intersection Problem for Tensors. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 32:1-32:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CCC.2026.32

Author Details

M. Levent Doğan
  • LMU Munich, Germany
John Maar
  • Technical University Berlin, Germany
Rafael Oliveira
  • University of Waterloo, Canada
Youming Qiao
  • University of Technology Sydney, Australia

Funding

  • Doğan, M. Levent: The author was funded by European Union (ERC Grant SYMOPTIC, 101040907) and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, 556164098).
  • Maar, John: The author was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, 556164098).
  • Qiao, Youming: Youming’s research is partly supported by ARC LP220100332.

Acknowledgements

The authors are grateful to Peter Bürgisser for valuable discussions during the Simons Program. This work started while the authors were supported by the Simons Institute for the Theory of Computing, and conducted when the authors were visiting the Institute. The authors would like to thank the generosity, hospitality, and excellent research environment provided by the Simons Institute.

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