Foundations of Computational Mathematics

, Volume 19, Issue 2, pp 245–258 | Cite as

Complexity Classes and Completeness in Algebraic Geometry

  • M. Umut IsikEmail author


We study the computational complexity of sequences of projective varieties. We define analogues of the complexity classes P and NP for these and prove the NP-completeness of a sequence called the universal circuit resultant. This is the first family of compact spaces shown to be NP-complete in a geometric setting.


Complexity classes Completeness Resultant 

Mathematics Subject Classification

14Q20 68Q15 



I would like to thank Vladimir Baranovsky and Saugata Basu for useful discussions on the subject of this paper.


  1. 1.
    Basu, S.: A complex analogue of Toda’s theorem. Foundations of Computational Mathematics 12(3), 327–362 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Basu, S.: A complexity theory of constructible functions and sheaves. Foundations of Computational Mathematics 15(1), 199–279 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Basu, S., Zell, T.: Polynomial hierarchy, Betti numbers, and a real analogue of Toda’s theorem. Foundations of Computational Mathematics 10(4), 429–454 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and real computation. Springer Science & Business Media (1998)Google Scholar
  5. 5.
    Bürgisser, P.: Completeness and reduction in algebraic complexity theory, vol. 7. Springer Science & Business Media (2013)Google Scholar
  6. 6.
    Bürgisser, P., Clausen, M., Shokrollahi, A.: Algebraic complexity theory, vol. 315. Springer Science & Business Media (2013)Google Scholar
  7. 7.
    Durand, A., Mahajan, M., Malod, G., de Rugy-Altherre, N., Saurabh, N.: Homomorphism polynomials complete for VP. In: LIPIcs-Leibniz International Proceedings in Informatics, vol. 29. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2014)Google Scholar
  8. 8.
    Gelfand, I.M., Kapranov, M., Zelevinsky, A.: Discriminants, resultants, and multidimensional determinants. Springer Science & Business Media (2008)Google Scholar
  9. 9.
    Kayal, N.: Affine projections of polynomials. In: Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pp. 643–662. ACM (2012)Google Scholar
  10. 10.
    Mahajan, M., Saurabh, N.: Some complete and intermediate polynomials in algebraic complexity theory. In: International Computer Science Symposium in Russia, pp. 251–265. Springer (2016)Google Scholar
  11. 11.
    Mulmuley, K.D.: On P vs. NP and geometric complexity theory: Dedicated to Sri Ramakrishna. Journal of the ACM (JACM) 58(2), 5 (2011)Google Scholar
  12. 12.
    Mulmuley, K.D., Sohoni, M.: Geometric complexity theory i: An approach to the P vs. NP and related problems. SIAM Journal on Computing 31(2), 496–526 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Raz, R.: Elusive functions and lower bounds for arithmetic circuits. In: Proceedings of the fortieth annual ACM symposium on Theory of computing, pp. 711–720. ACM (2008)Google Scholar
  14. 14.
    Shub, M.: Some problems for this century. Talk presented at FOCM14, December 11, 2014, Montevideo (2014). URL
  15. 15.
    Valiant, L.G.: Completeness classes in algebra. In: Proceedings of the eleventh annual ACM symposium on Theory of computing, pp. 249–261. ACM (1979)Google Scholar
  16. 16.
    Valiant, L.G.: The complexity of computing the permanent. Theoretical computer science 8(2), 189–201 (1979)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SFoCM 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, IrvineIrvineUSA

Personalised recommendations