VPSPACE and a Transfer Theorem over the Complex Field

  • Pascal Koiran
  • Sylvain Perifel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4708)


We extend the transfer theorem of [15] to the complex field. That is, we investigate the links between the class Open image in new window of families of polynomials and the Blum-Shub-Smale model of computation over Open image in new window . Roughly speaking, a family of polynomials is in Open image in new window if its coefficients can be computed in polynomial space. Our main result is that if (uniform, constant-free) Open image in new window families can be evaluated efficiently then the class Open image in new window of decision problems that can be solved in parallel polynomial time over the complex field collapses to Open image in new window . As a result, one must first be able to show that there are Open image in new window families which are hard to evaluate in order to separate Open image in new window from Open image in new window , or even from Open image in new window .


Polynomial Space Exponential Number Multivariate Polynomial Arithmetic Circuit Input Gate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, Heidelberg (1998)Google Scholar
  2. 2.
    Blum, L., Shub, M., Smale, S.: On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bulletin of the American Mathematical Society 21(1), 1–46 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bürgisser, P.: Completeness and Reduction in Algebraic Complexity Theory. Algorithms and Computation in Mathematics, vol. 7. Springer, Heidelberg (2000)zbMATHGoogle Scholar
  4. 4.
    Bürgisser, P.: On implications between P-NP-hypotheses: Decision versus computation in algebraic complexity. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 3–17. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  5. 5.
    Chapuis, O., Koiran, P.: Saturation and stability in the theory of computation over the reals. Annals of Pure and Applied Logic 99, 1–49 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Charbit, P., Jeandel, E., Koiran, P., Perifel, S., Thomassé, S.: Finding a vector orthogonal to roughly half a collection of vectors. Accepted in Journal of Complexity (2006), available from
  7. 7.
    Cole, R.: Parallel merge sort. SIAM J. Comput. 17(4), 770–785 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cucker, F., Grigoriev, D.: On the power of real Turing machines over binary inputs. SIAM J. Comput. 26(1), 243–254 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fitchas, N., Galligo, A., Morgenstern, J.: Precise sequential and parallel complexity bounds for quantifier elimination over algebraically closed fields. Journal of Pure and Applied Algebra 67, 1–14 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Grigoriev, D.: Topological complexity of the range searching. Journal of Complexity 16, 50–53 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Koiran, P.: Randomized and deterministic algorithms for the dimension of algebraic varieties. In: Proc. 38th FOCS, pp. 36–45 (1997)Google Scholar
  12. 12.
    Koiran, P.: Circuits versus trees in algebraic complexity. In: Reichel, H., Tison, S. (eds.) STACS 2000. LNCS, vol. 1770, pp. 35–52. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  13. 13.
    Koiran, P., Perifel, S.: Valiant’s model: from exponential sums to exponential products. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 596–607. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Koiran, P., Perifel, S.: VPSPACE and a transfer theorem over the complex field. Technical report, LIP, ENS Lyon (2007), Available from
  15. 15.
    Koiran, P., Perifel, S.: VPSPACE and a transfer theorem over the reals. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 417–428. Springer, Heidelberg (2007). Long version available from
  16. 16.
    Malod, G.: Polynômes et coefficients. PhD thesis, Université Claude Bernard Lyon 1 (July, 2003), available from
  17. 17.
    Malod, G., Portier, N.: Characterizing Valiant’s algebraic complexity classes. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 704–716. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  18. 18.
    Poizat, B.: Les petits cailloux. Aléas (1995)Google Scholar
  19. 19.
    Valiant, L.G.: Completeness classes in algebra. In: Proc. 11th ACM Symposium on Theory of Computing, pp. 249–261. ACM Press, New York (1979)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Pascal Koiran
    • 1
  • Sylvain Perifel
    • 1
  1. 1.LIP*, École Normale Supérieure de Lyon 

Personalised recommendations