License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2021.4
URN: urn:nbn:de:0030-drops-142781
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14278/
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Kumar, Mrinal ; Volk, Ben Lee

A Lower Bound on Determinantal Complexity

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LIPIcs-CCC-2021-4.pdf (0.7 MB)

Abstract

The determinantal complexity of a polynomial P ∈ ?[x₁, …, x_n] over a field ? is the dimension of the smallest matrix M whose entries are affine functions in ?[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.

BibTeX - Entry

@InProceedings{kumar_et_al:LIPIcs.CCC.2021.4,
  author =	{Kumar, Mrinal and Volk, Ben Lee},
  title =	{{A Lower Bound on Determinantal Complexity}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{4:1--4:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14278},
  URN =		{urn:nbn:de:0030-drops-142781},
  doi =		{10.4230/LIPIcs.CCC.2021.4},
  annote =	{Keywords: Determinantal Complexity, Algebraic Circuits, Lower Bounds, Singular Variety}
}

Keywords: Determinantal Complexity, Algebraic Circuits, Lower Bounds, Singular Variety
Collection: 36th Computational Complexity Conference (CCC 2021)
Issue Date: 2021
Date of publication: 08.07.2021

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