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A Lower Bound on Determinantal Complexity

Authors Mrinal Kumar, Ben Lee Volk

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Author Details

Mrinal Kumar
  • Department of Computer Science and Engineering, IIT Bombay, India
Ben Lee Volk
  • Department of Computer Science, University of Texas at Austin, TX, USA

Acknowledgements

Mrinal thanks Ramprasad Saptharishi for various discussions on determinantal complexity over the years, and in particular for explaining the proof of the result of Mignon and Ressayre to him.

Cite As Get BibTex

Mrinal Kumar and Ben Lee Volk. A Lower Bound on Determinantal Complexity. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 4:1-4:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.CCC.2021.4

Abstract

The determinantal complexity of a polynomial P ∈ 𝔽[x₁, …, x_n] over a field 𝔽 is the dimension of the smallest matrix M whose entries are affine functions in 𝔽[x₁, …, x_n] such that P = Det(M). We prove that the determinantal complexity of the polynomial ∑_{i = 1}^n x_i^n is at least 1.5n - 3. 
For every n-variate polynomial of degree d, the determinantal complexity is trivially at least d, and it is a long standing open problem to prove a lower bound which is super linear in max{n,d}. Our result is the first lower bound for any explicit polynomial which is bigger by a constant factor than max{n,d}, and improves upon the prior best bound of n + 1, proved by Alper, Bogart and Velasco [Jarod Alper et al., 2017] for the same polynomial.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Circuit complexity
Keywords
  • Determinantal Complexity
  • Algebraic Circuits
  • Lower Bounds
  • Singular Variety

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