Polynomial Time Algorithms in Invariant Theory for Torus Actions

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

Thumbnail PDF


  • Filesize: 0.82 MB
  • 30 pages

Document Identifiers

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

Cite AsGet BibTex

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


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Zeyuan Allen-Zhu, Ankit Garg, Yuanzhi Li, Rafael Oliveira, and Avi Wigderson. Operator scaling via geodesically convex optimization, invariant theory and polynomial identity testing. In STOC'18 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 172-181. ACM, New York, 2018. URL: https://doi.org/10.1145/3188745.3188942.
  2. Michele Audin. Torus actions on symplectic manifolds, volume 93. Birkhäuser, 2012. Google Scholar
  3. Eric Bach, James R. Driscoll, and Jeffrey O. Shallit. Factor refinement. In David S. Johnson, editor, Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, 22-24 January 1990, San Francisco, California, USA, pages 201-211. SIAM, 1990. URL: http://dl.acm.org/citation.cfm?id=320176.320199.
  4. Charles H Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K Wootters. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Physical review letters, 70(13):1895, 1993. Google Scholar
  5. Markus Bläser, Christian Ikenmeyer, Vladimir Lysikov, Anurag Pandey, and Frank-Olaf Schreyer. Variety membership testing, algebraic natural proofs, and geometric complexity theory. arXiv, 2020. URL: http://arxiv.org/abs/1911.02534.
  6. Peter A. Brooksbank and Eugene M. Luks. Testing isomorphism of modules. Journal of Algebra, 320(11):4020-4029, 2008. Computational Algebra. URL: https://doi.org/10.1016/j.jalgebra.2008.07.014.
  7. Peter Bürgisser, Matthias Christandl, Ketan D. Mulmuley, and Michael Walter. Membership in moment polytopes is in NP and coNP. SIAM J. Comput., 46(3):972-991, 2017. URL: https://doi.org/10.1137/15M1048859.
  8. Peter Bürgisser, Cole Franks, Ankit Garg, Rafael Mendes de Oliveira, Michael Walter, and Avi Wigderson. Towards a theory of non-commutative optimization: geodesic first and second order methods for moment maps and polytopes. In 60th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2019, pages 845-861. IEEE Computer Soc., Los Alamitos, CA, 2019. URL: http://arxiv.org/abs/1910.12375.
  9. Peter Bürgisser, Cole Franks, Ankit Garg, Rafael Oliveira, Michael Walter, and Avi Wigderson. Efficient algorithms for tensor scaling, quantum marginals, and moment polytopes. In 59th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2018, pages 883-897. IEEE Computer Soc., Los Alamitos, CA, 2018. URL: https://doi.org/10.1109/FOCS.2018.00088.
  10. Peter Bürgisser, Ankit Garg, Rafael Oliveira, Michael Walter, and Avi Wigderson. Alternating minimization, scaling algorithms, and the null-cone problem from invariant theory. In 9th Innovations in Theoretical Computer Science, volume 94 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 24, 20. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2018. Google Scholar
  11. Peter Bürgisser, Yinan Li, Harold Nieuwboer, and Michael Walter. Interior-point methods for unconstrained geometric programming and scaling problems. arXiv, 2020. URL: http://arxiv.org/abs/2008.12110.
  12. Michael B Cohen, Aleksander Madry, Dimitris Tsipras, and Adrian Vladu. Matrix scaling and balancing via box constrained Newton’s method and interior point methods. In Proceedings of the Symposium on Foundations of Computer Science (FOCS 2017), pages 902-913. IEEE, 2017. URL: http://arxiv.org/abs/1704.02310.
  13. Stephen A. Cook. The complexity of theorem proving procedures. In Proc. 3rd ACM STOC, pages 151-158, 1971. Google Scholar
  14. David A. Cox, John Little, and Donal O'Shea. Ideals, varieties, and algorithms - an introduction to computational algebraic geometry and commutative algebra (2. ed.). Undergraduate texts in mathematics. Springer, 1997. Google Scholar
  15. Harm Derksen. Polynomial bounds for rings of invariants. Proc. Amer. Math. Soc., 129(4):955-963, 2001. URL: https://doi.org/10.1090/S0002-9939-00-05698-7.
  16. Harm Derksen. The graph isomorphism problem and approximate categories. J. Symb. Comput., 59:81-112, 2013. URL: https://doi.org/10.1016/j.jsc.2013.06.002.
  17. Harm Derksen and Gregor Kemper. Computational invariant theory, volume 130 of Encyclopaedia of Mathematical Sciences. Springer, Heidelberg, enlarged edition, 2015. With two appendices by Vladimir L. Popov, and an addendum by Norbert A'Campo and Popov, Invariant Theory and Algebraic Transformation Groups, VIII. URL: https://doi.org/10.1007/978-3-662-48422-7.
  18. Harm Derksen and Visu Makam. Generating invariant rings of quivers in arbitrary characteristic. J. Algebra, 489:435-445, 2017. URL: https://doi.org/10.1016/j.jalgebra.2017.06.035.
  19. Harm Derksen and Visu Makam. Polynomial degree bounds for matrix semi-invariants. Adv. Math., 310:44-63, 2017. URL: https://doi.org/10.1016/j.aim.2017.01.018.
  20. Harm Derksen and Visu Makam. Degree bounds for semi-invariant rings of quivers. J. Pure Appl. Algebra, 222(10):3282-3292, 2018. URL: https://doi.org/10.1016/j.jpaa.2017.12.007.
  21. Harm Derksen and Visu Makam. Algorithms for orbit closure separation for invariants and semi-invariants of matrices. Algebra Number Theory, 14(10):2791-2813, 2020. URL: https://doi.org/10.2140/ant.2020.14.2791.
  22. Harm Derksen and Visu Makam. An exponential lower bound for the degrees of invariants of cubic forms and tensor actions. Adv. Math., 368:107136, 25, 2020. URL: https://doi.org/10.1016/j.aim.2020.107136.
  23. Igor Dolgachev. Lectures on invariant theory, volume 296 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2003. URL: https://doi.org/10.1017/CBO9780511615436.
  24. Arnaud Durand, Miki Hermann, and Laurent Juban. On the complexity of recognizing the Hilbert basis of a linear Diophantine system. Theoret. Comput. Sci., 270(1-2):625-642, 2002. URL: https://doi.org/10.1016/S0304-3975(01)00017-2.
  25. Kousha Etessami, Alistair Stewart, and Mihalis Yannakakis. A note on the complexity of comparing succinctly represented integers, with an application to maximum probability parsing. ACM Trans. Comput. Theory, 6(2):9:1-9:23, 2014. URL: https://doi.org/10.1145/2601327.
  26. Michael A. Forbes and Amir Shpilka. Explicit Noether normalization for simultaneous conjugation via polynomial identity testing. In Approximation, randomization, and combinatorial optimization, volume 8096 of Lecture Notes in Comput. Sci., pages 527-542. Springer, Heidelberg, 2013. URL: https://doi.org/10.1007/978-3-642-40328-6_37.
  27. Ankit Garg, Leonid Gurvits, Rafael Oliveira, and Avi Wigderson. A deterministic polynomial time algorithm for non-commutative rational identity testing. In 57th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2016, pages 109-117. IEEE Computer Soc., Los Alamitos, CA, 2016. URL: https://doi.org/10.1109/FOCS.2016.95.
  28. Ankit Garg, Leonid Gurvits, Rafael Oliveira, and Avi Wigderson. Operator scaling: theory and applications. Found. Comput. Math., 20(2):223-290, 2020. URL: https://doi.org/10.1007/s10208-019-09417-z.
  29. Ankit Garg, Christian Ikenmeyer, Visu Makam, Rafael Mendes de Oliveira, Michael Walter, and Avi Wigderson. Search problems in algebraic complexity, GCT, and hardness of generators for invariant rings. In Shubhangi Saraf, editor, 35th Computational Complexity Conference, CCC 2020, July 28-31, 2020, Saarbrücken, Germany (Virtual Conference), volume 169 of LIPIcs, pages 12:1-12:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.12.
  30. Guoqiang Ge. Testing equalities of multiplicative representations in polynomial time (extended abstract). In 34th Annual Symposium on Foundations of Computer Science, Palo Alto, California, USA, 3-5 November 1993, pages 422-426. IEEE Computer Society, 1993. URL: https://doi.org/10.1109/SFCS.1993.366845.
  31. Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric algorithms and combinatorial optimization, volume 2 of Algorithms and Combinatorics. Springer-Verlag, Berlin, second edition, 1993. URL: https://doi.org/10.1007/978-3-642-78240-4.
  32. Victor Guillemin and Shlomo Sternberg. Symplectic techniques in physics. Cambridge university press, 1990. Google Scholar
  33. Leonid Gurvits. Classical complexity and quantum entanglement. J. Comput. Syst. Sci., 69(3):448-484, 2004. URL: https://doi.org/10.1016/j.jcss.2004.06.003.
  34. Leonid Gurvits. Combinatorial and algorithmic aspects of hyperbolic polynomials. arXiv preprint, 2004. URL: http://arxiv.org/abs/math/0404474.
  35. D. Hilbert. Über die vollen Invariantensysteme. Math. Ann., 42:313-373, 1893. URL: http://eudml.org/doc/157652.
  36. David Hilbert. Über die Theorie der algebraischen Formen. Math. Ann., 36(4):473-534, 1890. URL: https://doi.org/10.1007/BF01208503.
  37. Evelyne Hubert and George Labahn. Scaling invariants and symmetry reduction of dynamical systems. Foundations of Computational Mathematics, 13, 2013. URL: https://doi.org/10.1007/s10208-013-9165-9.
  38. James E. Humphreys. Linear algebraic groups. Springer-Verlag, New York-Heidelberg, 1975. Graduate Texts in Mathematics, No. 21. Google Scholar
  39. Christian Ikenmeyer, Ketan D Mulmuley, and Michael Walter. On vanishing of Kronecker coefficients. Computational Complexity, 26(4):949-992, 2017. Google Scholar
  40. Gábor Ivanyos, Youming Qiao, and K. V. Subrahmanyam. Non-commutative Edmonds' problem and matrix semi-invariants. Comput. Complexity, 26(3):717-763, 2017. URL: https://doi.org/10.1007/s00037-016-0143-x.
  41. Gábor Ivanyos, Youming Qiao, and K. V. Subrahmanyam. Constructive non-commutative rank computation is in deterministic polynomial time. Comput. Complexity, 27(4):561-593, 2018. URL: https://doi.org/10.1007/s00037-018-0165-7.
  42. Valentine Kabanets and Russell Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity, 13(1-2):1-46, 2004. URL: https://doi.org/10.1007/s00037-004-0182-6.
  43. Ravindran Kannan and Achim Bachem. Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix. SIAM J. Comput., 8(4):499-507, 1979. URL: https://doi.org/10.1137/0208040.
  44. Richard M. Karp. Reducibility among combinatorial problems. In Complexity of computer computations (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 1972), pages 85-103, 1972. Google Scholar
  45. George Kempf and Linda Ness. The length of vectors in representation spaces. In Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), volume 732 of Lecture Notes in Math., pages 233-243. Springer, Berlin, 1979. Google Scholar
  46. Hanspeter Kraft. Geometrische Methoden in der Invariantentheorie. Aspects of Mathematics, D1. Friedr. Vieweg & Sohn, Braunschweig, 1984. URL: https://doi.org/10.1007/978-3-322-83813-1.
  47. David B Leep and Gerry Myerson. Marriage, magic, and solitaire. The American Mathematical Monthly, 106(5):419-429, 1999. Google Scholar
  48. L. A. Levin. Universal enumeration problems. Problemy Peredači Informacii, 9(3):115-116, 1973. Google Scholar
  49. Nathan Linial and Zur Luria. On the vertices of the d-dimensional Birkhoff polytope. Discrete & Computational Geometry, 51(1):161-170, 2014. Google Scholar
  50. Nathan Linial, Alex Samorodnitsky, and Avi Wigderson. A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents. Combinatorica, 20(4):545-568, 2000. Google Scholar
  51. Visu Makam and Avi Wigderson. Singular tuples of matrices is not a null cone (and, the symmetries of algebraic varieties). CoRR, abs/1909.00857, 2019. URL: http://arxiv.org/abs/1909.00857.
  52. Ketan D. Mulmuley. Geometric complexity theory V: Efficient algorithms for Noether normalization. J. Amer. Math. Soc., 30(1):225-309, 2017. URL: https://doi.org/10.1090/jams/864.
  53. Ketan D Mulmuley and Milind Sohoni. Geometric complexity theory I: An approach to the P vs. NP and related problems. SIAM Journal on Computing, 31(2):496-526, 2001. Google Scholar
  54. David Mumford. The red book of varieties and schemes, volume 1358 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1988. URL: https://doi.org/10.1007/978-3-662-21581-4.
  55. David Mumford, John Fogarty, and Frances Kirwan. Geometric invariant theory, Third Edition, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, 1994. Google Scholar
  56. Vladimir L Popov. Two orbits: When is one in the closure of the other? Proceedings of the Steklov Institute of Mathematics, 264(1):146-158, 2009. Google Scholar
  57. Alexander Schrijver. Theory of linear and integer programming. Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons, Ltd., Chichester, 1986. A Wiley-Interscience Publication. Google Scholar
  58. R. Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. The Annals of Mathematical Statistics, 35:876-879, 1964. Google Scholar
  59. Henry J. Stephen Smith. On systems of linear indeterminate equations and congruences. Philosophical Transactions of the Royal Society of London, 151:293-326, 1861. URL: http://www.jstor.org/stable/108738.
  60. Bernd Sturmfels. Algorithms in Invariant Theory. Texts & Monographs in Symbolic Computation. Springer, 2008. URL: https://doi.org/10.1007/978-3-211-77417-5.
  61. David Wehlau. Constructive invariant theory for tori. Annales de l'institut Fourier, 43(4):1055-1066, 1993. URL: http://eudml.org/doc/75025.
  62. Hermann Weyl. The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J., 1939. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail