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Search Problems in Algebraic Complexity, GCT, and Hardness of Generators for Invariant Rings

Authors Ankit Garg, Christian Ikenmeyer, Visu Makam, Rafael Oliveira, Michael Walter, Avi Wigderson

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Author Details

Ankit Garg
  • Microsoft Research, Bangalore, India
Christian Ikenmeyer
  • University of Liverpool, UK
Visu Makam
  • Institute for Advanced Study, Princeton, NJ, USA
Rafael Oliveira
  • University of Waterloo, Canada
Michael Walter
  • Korteweg-de Vries Institute for Mathematics, Institute for Theoretical Physics, Institute for Logic, Language &Computation, University of Amsterdam, The Netherlands
Avi Wigderson
  • Institute for Advanced Study, Princeton, NJ, US

Funding

  • Ikenmeyer, Christian: DFG grant IK 116/2-1.
  • Makam, Visu: NSF grant No. DMS -1638352 and NSF grant No. CCF-1412958.
  • Walter, Michael: NWO Veni grant 680-47-459.
  • Wigderson, Avi: NSF grant No. CCF-1412958 and NSF grant CCF-1900460.

Cite As Get BibTex

Ankit Garg, Christian Ikenmeyer, Visu Makam, Rafael Oliveira, Michael Walter, and Avi Wigderson. Search Problems in Algebraic Complexity, GCT, and Hardness of Generators for Invariant Rings. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.CCC.2020.12

Abstract

We consider the problem of computing succinct encodings of lists of generators for invariant rings for group actions. Mulmuley conjectured that there are always polynomial sized such encodings for invariant rings of SL_n(ℂ)-representations. We provide simple examples that disprove this conjecture (under standard complexity assumptions).
We develop a general framework, denoted algebraic circuit search problems, that captures many important problems in algebraic complexity and computational invariant theory. This framework encompasses various proof systems in proof complexity and some of the central problems in invariant theory as exposed by the Geometric Complexity Theory (GCT) program, including the aforementioned problem of computing succinct encodings for generators for invariant rings.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • generators for invariant rings
  • succinct encodings

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