Skip to main content
Log in

Finding connected components of a semialgebraic set in subexponential time

  • Published:
Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

Let a semialgebraic set be given by a quantifier-free formula of the first-order theory of real closed fields with atomic subformulae of type (f i ≥ 0), 1 ≤i ≤k where the polynomialsf i ε ℤ[X 1,..., Xn] have degrees deg(f i <d and the absolute value of each (integer) coefficient off i is at most 2M. An algorithm is designed which finds the connected components of the semialgebraic set in time\(M^{O(1)} (kd)^{n^{O(1)} } \). The best previously known bound\(M^{O(1)} (kd)^{n^{O(n)} } \) for this problem follows from Collins' method of Cylindrical Algebraic Decomposition.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Grigor'ev, D.Yu., Vorobjov, N.N., jr.: Solving systems of polynomial inequalities in subexponential time. J. Symb. Comput.5, 37–64 (1988)

    Google Scholar 

  2. Grigor'ev, D.Yu.: Complexity of deciding Tarski algebra. J. Symb. Comput.5, 65–108 (1988)

    Google Scholar 

  3. Tarski, A.: A desision method for elementary algebra and geometry. University of California Press 1951

  4. Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. Lecture Notes Comp. Sci. Vol. 33, pp. 134–183. Berlin, Heidelberg, New York: Springer 1975

    Google Scholar 

  5. Wüthrich, H.R.: Ein Entscheidungsverfahren für die Theorie der reell-abgeschlossenen Körper. Lecture Notes Comp. Sci. Vol. 43, pp. 138–162. Berlin, Heidelberg, New York: Springer 1976

    Google Scholar 

  6. Chistov, A.L., Grigor'ev, D.Yu.: Subexponential-time solving systems of algebraic equations. I. II. Preprints LOMI E-9-83 and E-10-83, Leningrad 1983

  7. Grigor'ev, D.Yu.: Computational complexity in polynomial algebra. Proceedings of the 29th Int. Congress of Mathematicians. Berkeley, pp. 1452–1460 (1986)

  8. Lang, S.: Algebra. New York: Addison-Wesley 1965

    Google Scholar 

  9. Fitchos, N., Galligo, A., Morgenstern, J.: Algorithmes rapides en séquential et en parallele pour l'élimination de quantificateurs en géométrie élémentaire. VER de Mathématiques Université de Paris VII (1988)

  10. Vorobjov, N.N., jr.: Deciding consistency of systems of polynomial in exponent inequalities in subexponejitial time. Notes of Sci. Seminars of Leningrad Department of Math. Steklov Inst.176, 3–52 (in Russian) (1989)

    Google Scholar 

  11. Shafarevich, I.R.: Basic algebraic geometry. Berlin, Heidelberg, New York: Springer 1974

    Google Scholar 

  12. Heintz, J.: Definability and fast quantifier elimination in algebraically closed field. Theor. Comp. Sci.24, 239–278 (1983)

    Google Scholar 

  13. Vorobjov, N.N., jr., Grigor'ev, D.Yu.: Counting connected components of a semialgebraic set in subexponential time. Sov. Math. Dokl.42(2), 563–566 (1991)

    Google Scholar 

  14. Dold, A.: Lectures on algebraic topology. Berlin, Heidelberg, New York: Springer 1972

    Google Scholar 

  15. Grigor'ev, D.Yu., Vorobjov, N.N., jr.: Counting connected components of a semialgebraic set in subexponential time. Submitted to Computational Complexity 1989

  16. Heintz, J., Roy, M.-F., Solerno, P.: Sur la complexité du principe de Tarski-Seidenberg. Bull. Soc. Math. France.118, 101–126 (1990)

    Google Scholar 

  17. Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals. Parts I-II. Technical report, Cornell University 1989

  18. Canny, J.F.: The complexity of robot motion planning. Cambridge: MIT Press 1988

    Google Scholar 

  19. Grigor'ev, D.Yu., Heintz, J., Roy, M.-F., Solerno, P., Vorobjov, N.N., jr.: Comptage des composantes connexes d'un ensemble semi-algebrique en temps simplement exponential. CompteRendus Acad. Sci. Paris311, Serié I, 879–822 (1990)

    Google Scholar 

  20. Heintz, J., Roy, M.-F., Solerno, P.: Construction de chamins dans un ensemble semi-algebrique. Preprint Univ. Buenos Aires, Argentina 1990

    Google Scholar 

  21. Heintz, J., Roy, M.-F., Solerno, P.: Single exponential path finding in semialgebraic sets. Proc. AAECC Conf. Tokyo 1990

  22. Heintz, J., Roy, M.-F., Solerno, P.: Description des composantes connexes d'un ensemble semi-algebrique en temps simplement exponentiel. Compte-Rendus Acad. Sci. Paris313, Serié I, 167–170 (1991)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Canny, J., Grigor'ev, D.Y. & Vorobjov, N.N. Finding connected components of a semialgebraic set in subexponential time. AAECC 2, 217–238 (1992). https://doi.org/10.1007/BF01614146

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01614146

Key words

Navigation