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Polynomial Time Algorithms in Invariant Theory for Torus Actions

Authors Peter Bürgisser, M. Levent Doğan, Visu Makam, Michael Walter, Avi Wigderson

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Author Details

Peter Bürgisser
  • Institut für Mathematik, Technische Universität Berlin, Germany
M. Levent Doğan
  • Institut für Mathematik, Technische Universität Berlin, Germany
Visu Makam
  • Institute for Advanced Study, Princeton, NJ, CA
  • School of Mathematics and Statistics, University of Melbourne, Australia
Michael Walter
  • Korteweg-de Vries Institute for Mathematics, Institute for Theoretical Physics, Institute for Logic, Language and Computation, University of Amsterdam, The Netherlands
Avi Wigderson
  • Institute for Advanced Study, Princeton, NJ, USA

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Peter Bürgisser, M. Levent Doğan, Visu Makam, Michael Walter, and Avi Wigderson. Polynomial Time Algorithms in Invariant Theory for Torus Actions. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 32:1-32:30, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


An action of a group on a vector space partitions the latter into a set of orbits. We consider three natural and useful algorithmic "isomorphism" or "classification" problems, namely, orbit equality, orbit closure intersection, and orbit closure containment. These capture and relate to a variety of problems within mathematics, physics and computer science, optimization and statistics. These orbit problems extend the more basic null cone problem, whose algorithmic complexity has seen significant progress in recent years. In this paper, we initiate a study of these problems by focusing on the actions of commutative groups (namely, tori). We explain how this setting is motivated from questions in algebraic complexity, and is still rich enough to capture interesting combinatorial algorithmic problems. While the structural theory of commutative actions is well understood, no general efficient algorithms were known for the aforementioned problems. Our main results are polynomial time algorithms for all three problems. We also show how to efficiently find separating invariants for orbits, and how to compute systems of generating rational invariants for these actions (in contrast, for polynomial invariants the latter is known to be hard). Our techniques are based on a combination of fundamental results in invariant theory, linear programming, and algorithmic lattice theory.

Subject Classification

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


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