Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Recent Advances in the Computation of the Homology of Semialgebraic Sets

  • Conference paper
  • First Online:
Computing with Foresight and Industry (CiE 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11558))

Included in the following conference series:

Abstract

This article describes recent advances in the computation of the homology groups of semialgebraic sets. It summarizes a series of papers by the author and several coauthors (P. Bürgisser, T. Krick, P. Lairez, M. Shub, and J. Tonelli-Cueto) on which a sequence of ideas and techniques were deployed to tackle the problem at increasing levels of generality. The goal is not to provide a detailed technical picture but rather to throw light on the main features of this technical picture, the complexity results obtained, and how the new algorithms fit into the landscape of existing results.

Partially supported by a GRF grant from the Research Grants Council of the Hong Kong SAR (project number CityU 11202017).

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Amelunxen, D., Lotz, M.: Average-case complexity without the black swans. J. Complexity 41, 82–101 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Basu, S.: On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets. In: Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996), pp. 408–417. ACM, New York (1996)

    Google Scholar 

  3. Basu, S.: Computing the first few Betti numbers of semi-algebraic sets in single exponential time. J. Symbolic Comput. 41(10), 1125–1154 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Basu, S., Pollack, R., Roy, M.F.: On the combinatorial and algebraic complexity of quantifier elimination. J. ACM 43, 1002–1045 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Basu, S., Pollack, R., Roy, M.F.: Computing roadmaps of semi-algebraic sets on a variety. J. AMS 33, 55–82 (1999)

    MATH  Google Scholar 

  6. Basu, S., Pollack, R., Roy, M.F.: Algorithms in real algebraic geometry. Algorithms and Computation in Mathematics, vol. 10, 2nd edn. Springer, Berlin (2006)

    Book  MATH  Google Scholar 

  7. Benedetti, R., Risler, J.J.: Real Algebraic and Semi-algebraic Sets, Hermann (1990)

    Google Scholar 

  8. Bochnak, J., Coste, M., Roy, M.F.: Real Algebraic Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics, vol. 36. Springer, Heidelberg (1998). https://doi.org/10.1007/978-3-662-03718-8. Translated from the 1987 French original, Revised by the authors

    Book  MATH  Google Scholar 

  9. Bürgisser, P., Cucker, F.: Condition. Grundlehren der mathematischen Wissenschaften, vol. 349. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38896-5

    Book  MATH  Google Scholar 

  10. Bürgisser, P., Cucker, F., Lairez, P.: Computing the homology of basic semialgebraic sets in weakly exponential time. J. ACM 66(1), 5:1–5:30 (2019)

    Article  MATH  Google Scholar 

  11. Bürgisser, P., Cucker, F., Lotz, M.: The probability that a slightly perturbed numerical analysis problem is difficult. Math. Comput. 77, 1559–1583 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bürgisser, P., Cucker, F., Tonelli-Cueto, J.: Computing the homology of semialgebraic sets. I: Lax formulas. Found. Comput. Math. arXiv:1807.06435 (2018)

  13. Bürgisser, P., Cucker, F., Tonelli-Cueto, J.: Computing the homology of semialgebraic sets. II: Arbitrary formulas (2019). preprint

    Google Scholar 

  14. Canny, J.: Computing roadmaps of general semi-algebraic sets. Comput. J. 36(5), 504–514 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Canny, J., Grigorev, D., Vorobjov, N.: Finding connected components of a semialgebraic set in subexponential time. Appl. Algebra Eng. Commun. Comput. 2(4), 217–238 (1992)

    Article  MathSciNet  Google Scholar 

  16. Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975). https://doi.org/10.1007/3-540-07407-4_17

    Chapter  Google Scholar 

  17. Cucker, F.: Approximate zeros and condition numbers. J. Complexity 15, 214–226 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cucker, F., Krick, T., Shub, M.: Computing the homology of real projective sets. Found. Comp. Math. 18, 929–970 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cucker, F., Smale, S.: Complexity estimates depending on condition and round-off error. J. ACM 46, 113–184 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Demmel, J.: The probability that a numerical analysis problem is difficult. Math. Comput. 50, 449–480 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gabrielov, A., Vorobjov, N.: Approximation of definable sets by compact families, and upper bounds on homotopy and homology. J. Lond. Math. Soc. 80(1), 35–54 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Goldstine, H., von Neumann, J.: Numerical inverting matrices of high order, II. Proc. AMS 2, 188–202 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  23. Grigoriev, D.: Complexity of deciding Tarski algebra. J. Symbolic Comput. 5, 65–108 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Grigoriev, D., Vorobjov, N.: Solving systems of polynomial inequalities in subexponential time. J. Symbolic Comput. 5, 37–64 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  25. Grigoriev, D., Vorobjov, N.: Counting connected components of a semialgebraic set in subexponential time. Comput. Complexity 2, 133–186 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  27. Heintz, J., Roy, M.F., Solerno, P.: Single exponential path finding in semi-algebraic sets II: the general case. In: Bajaj, C. (ed.) Algebraic Geometry and its Applications, pp. 449–465. Springer, New York (1994). https://doi.org/10.1007/978-1-4612-2628-4_28

    Chapter  MATH  Google Scholar 

  28. Koiran, P.: The real dimension problem is \({\sf N}P_{\mathbb{R}}\)-complete. J. Complexity 15, 227–238 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kostlan, E.: Complexity theory of numerical linear algebra. J. Comput. Appl. Math. 22, 219–230 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  30. von Neumann, J., Goldstine, H.: Numerical inverting matrices of high order. Bull. AMS 53, 1021–1099 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  31. Niyogi, P., Smale, S., Weinberger, S.: Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom. 39, 419–441 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals. Part I. J. Symbolic Comput. 13, 255–299 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  33. Renegar, J.: Some perturbation theory for linear programming. Math. Program. 65, 73–91 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  34. Renegar, J.: Incorporating condition measures into the complexity theory of linear programming. SIAM J. Optim. 5, 506–524 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  35. Renegar, J.: Linear programming, complexity theory and elementary functional analysis. Math. Program. 70, 279–351 (1995)

    MathSciNet  MATH  Google Scholar 

  36. Shub, M., Smale, S.: Complexity of Bézout’s theorem I: geometric aspects. J. AMS 6, 459–501 (1993)

    MATH  Google Scholar 

  37. Smale, S.: Complexity theory and numerical analysis. In: Iserles, A. (ed.) Acta Numerica, pp. 523–551. Cambridge University Press (1997)

    Google Scholar 

  38. Spielman, D., Teng, S.H.: Smoothed analysis: an attempt to explain the behavior of algorithms in practice. Commun. ACM 52(10), 77–84 (2009)

    Article  Google Scholar 

  39. Tarski, A.: A Decision Method for Elementary Algebra and Geometry. University of California Press, Berkeley (1951)

    MATH  Google Scholar 

  40. Turing, A.: On computable numbers, with an application to the Entscheidungsproblem. Proc. London Math. Soc. S2–42, 230–265 (1936)

    MathSciNet  MATH  Google Scholar 

  41. Turing, A.: Rounding-off errors in matrix processes. Quart. J. Mech. Appl. Math. 1, 287–308 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wilkinson, J.: Some comments from a numerical analyst. J. ACM 18, 137–147 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wüthrich, H.R.: Ein Entscheidungsverfahren für die Theorie der reell-abgeschlossenen Körper. In: Strassen, V., Specker, E. (eds.) Komplexität von Entscheidungsproblemen Ein Seminar. LNCS, vol. 43, pp. 138–162. Springer, Heidelberg (1976). https://doi.org/10.1007/3-540-07805-3_10

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong

    Felipe Cucker

Authors
  1. Felipe Cucker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Felipe Cucker .

Editor information

Editors and Affiliations

  1. Christian Albrechts University of Kiel, Kiel, Germany

    Florin Manea

  2. Durham University, Durham, UK

    Barnaby Martin

  3. Durham University, Durham, UK

    Daniël Paulusma

  4. Università degli Studi di Milano, Milan, Italy

    Giuseppe Primiero

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Cucker, F. (2019). Recent Advances in the Computation of the Homology of Semialgebraic Sets. In: Manea, F., Martin, B., Paulusma, D., Primiero, G. (eds) Computing with Foresight and Industry. CiE 2019. Lecture Notes in Computer Science(), vol 11558. Springer, Cham. https://doi.org/10.1007/978-3-030-22996-2_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-22996-2_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-22995-5

  • Online ISBN: 978-3-030-22996-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation