An Efficient Algorithm for the Sign Condition Problem in the Semi-algebraic Context

  • Rafael Grimson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6130)


We study algebraic complexity of the sign condition problem for any given family of polynomials. Essentially, the problem consists in determining the sign condition satisfied by a fixed family of polynomials at a query point, performing as little arithmetic operations as possible. After defining precisely the sign condition and the point location problems, we introduce a method called the dialytic method to solve the first problem efficiently. This method involves a linearization of the original polynomials and provides the best known algorithm to solve the sign condition problem. Moreover, we prove a lower bound showing that the dialytic method is almost optimal. Finally, we extend our method to the point location problem.

The dialytic method solves (non-uniformly) the sign condition problem for a family of s polynomials in R[X 1,...,X n ] given by an arithmetic circuit \(\Gamma_{\mathcal F}\) of non-scalar complexity L performing \({\mathcal O}((L+n)^5\log(s))\) arithmetic operations.

If the polynomials are given in dense representation and d is a bound for their degrees, the complexity of our method is \({\mathcal O}(d^{5n} log(s))\). Comparable bounds are obtained for the point location problem.


Arithmetic Operation Query Point Query Time Dialytic Method Arithmetic Circuit 
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.


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  1. [BCR98]
    Bochnak, J., Coste, M., Roy, M.F.: Real Algebraic Geometry. Modern Surveys in Mathematics, vol. 36. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  2. [BCS97]
    Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic complexity theory. Grundlehren der mathematischen Wissenschaften. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  3. [BH99]
    Baur, W., Halupczok, H.: On lower bounds for the complexity of polynomials and their multiples. Computational Complexity 8(4), 309–315 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  4. [BO83]
    Ben-Or, M.: Lower bounds for algebraic computation trees. In: STOC 1983: Proceedings of the fifteenth annual ACM symposium on Theory of computing, pp. 80–86. ACM, New York (1983)CrossRefGoogle Scholar
  5. [BPR10]
    Basu, S., Pollack, R., Roy, M.F.: An asymptotically tight bound on the number of connected components of realizable sign conditions. Combinatorica (2009/2010) (to appear)Google Scholar
  6. [BPR99]
    Basu, S., Pollack, R., Roy, M.-F.: Computing roadmaps of semi-algebraic sets on a variety. J. AMS 3(1), 55–82 (1999)MathSciNetGoogle Scholar
  7. [BPR06]
    Basu, S., Pollack, R., Roy, M.F.: Algorithms in real algebraic geometry. In: Algorithms and Computation in Mathematics, 2nd edn., vol. 10. Springer, Secaucus (2006)Google Scholar
  8. [Can93]
    Canny, J.F.: Computing roadmaps of general semi-algebraic sets. Comput. J. 36(5), 504–514 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  9. [CEGS91]
    Chazelle, B., Edelsbrunner, H., Guibas, L.J., Sharir, M.: A singly-exponential stratification scheme for real semi-algebraic varieties and its applications. Theoretical Computer Science 84(1), 77–105 (1991)zbMATHCrossRefGoogle Scholar
  10. [CGV91]
    Canny, J.F., Grigoriev, D., Vorobjov, N.: Finding connected components of a simialgebraic set in subexponential time. Appl. Algebra Eng. Commun. Comput. 2, 217–238 (1991)CrossRefGoogle Scholar
  11. [Cla87]
    Clarkson, K.L.: New applications of random sampling in computational geometry. Discrete & Computational Geometry 2(2), 195–222 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  12. [Col75]
    Collins, G.E.: Hauptvortrag: Quantifier elimination for real closed fields by Cylindrical Algebraic Decomposition. Automata Theory and Formal Languages, 134–183 (1975)Google Scholar
  13. [CS90]
    Chazelle, B., Sharir, M.: An algorithm for generalized point location and its applications. J. Symb. Computation 10(3-4), 281–309 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  14. [DL76]
    Dobkin, D.P., Lipton, R.J.: Multidimensional searching problems. SIAM J. Comput. 5(2), 181–186 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  15. [GHR+90]
    Grigoriev, D., Heintz, J., Roy, M.F., Solern, P., Vorobjov, N.: Comptage des composantes connexes d’un ensemble semi-algbrique en temps simplement exponentiel. C.R. Acad. Sci. Paris 311, 879–882 (1990)Google Scholar
  16. [GR93]
    Gournay, L., Risler, J.J.: Construction of roadmaps in semi-algebraic sets. Appl. Algebra Eng. Commun. Comput. 4, 239–252 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  17. [Gri88]
    Grigoriev, D.: Complexity of deciding tarski algebra. J. Symb. Computation 5(1/2), 65–108 (1988)CrossRefGoogle Scholar
  18. [Gri00]
    Grigoriev, D.: Topological complexity of the range searching. J. Complexity 16(1), 50–53 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  19. [GV92]
    Grigoriev, D., Vorobjov, N.: Counting connected components of a semialgebraic set in subexponential time. Computational Complexity 2, 133–186 (1992)CrossRefMathSciNetGoogle Scholar
  20. [HRS90a]
    Heintz, J., Roy, M.F., Solern, P.: Single exponential path finding in semialgebraic sets. Part 1: The case of a regular bounded hypersurface. In: Sakata, S. (ed.) AAECC 1990. LNCS, vol. 508, pp. 180–196. Springer, Heidelberg (1991)Google Scholar
  21. [HRS90b]
    Heintz, J., Roy, M.F., Solern, P.: Sur la complexit du principe de Tarski-Seidenberg. Bull. SMF 118, 101–126 (1990)zbMATHGoogle Scholar
  22. [HRS94a]
    Heintz, J., Roy, M.F., Solern, P.: Description of the connected components of a semialgebraic in single exponential time. Discrete & Computational Geometry 11, 121–140 (1994)Google Scholar
  23. [HRS94b]
    Heintz, J., Roy, M.F., Solern, P.: Single exponential path finding in semi-algebraic sets II: The general case, Algebraic geometry and its applications, collections of papers from Abhyankar’s 60-th birthday conference, Purdue University, West-Lafayette (1994)Google Scholar
  24. [JS00]
    Jeronimo, G., Sabia, J.: On the number of sets definable by polynomials. J. of Algebra 227(2), 633–644 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  25. [Kha79]
    Khachiyan, L.G.: A polynomial algorithm in linear programming. Soviet Mathematics Doklady 20, 191–194 (1979)zbMATHGoogle Scholar
  26. [Koi00]
    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
  27. [LB01]
    Ganapathy, M.K., Babai, L., Ranyai, L.: On the number of zero-patterns of a sequence of polynomials. J. American Math. Soc. 14, 717–735 (2001)zbMATHCrossRefGoogle Scholar
  28. [Lic90]
    Lickteig, T.: On semialgebraic decision complexity, Technical Report TR-90-052, Int. Comput. Sc. Inst, Berkeley. Habilitationsschrift, Universitat Tubingen (1990)Google Scholar
  29. [LSJM40]
    Liddell, H.G., Scott, R., Jones, H.S., Mckenzie, R.: A greek-english lexicon. Clarendon Press, Oxford (1940)Google Scholar
  30. [MadH84]
    Meyer auf der Heide, F.: A polynomial linear search algorithm for the n-dimensional knapsack problem. J. ACM 31(3), 668–676 (1984)zbMATHMathSciNetGoogle Scholar
  31. [MadH88]
    Meyer auf der Heide, F.: Fast algorithms for n-dimensional restrictions of hard problems. J. ACM 35(3), 740–747 (1988)Google Scholar
  32. [Mei93]
    Meiser, S.: Point location in arrangements of hyperplanes. Inf. Comput. 106(2), 286–303 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  33. [Mil64]
    Milnor, J.: On the Betti numbers of real varieties. Proc. AMS 15, 275–280 (1964)zbMATHCrossRefMathSciNetGoogle Scholar
  34. [Sno04]
    Snoeyink, J.: Point location. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry. CRC Press LLC, Boca Raton (2004)Google Scholar
  35. [Str81]
    Strassen, V.: The computational complexity of continued fractions. In: SYMSAC 1981: Proceedings of the fourth ACM symposium on Symbolic and algebraic computation, pp. 51–67. ACM, New York (1981)CrossRefGoogle Scholar
  36. [Syl42]
    Sylvester, J.J.: Memoir on the dialytic method of elimination. Part I. Philosophical Magazine XXI, 534–539 (1842)Google Scholar
  37. [vzG86]
    von zur Gathen, J.: Parallel arithmetic computations: a survey. In: Wiedermann, J., Gruska, J., Rovan, B. (eds.) MFCS 1986. LNCS, vol. 233, pp. 93–112. Springer, Heidelberg (1986)CrossRefGoogle Scholar
  38. [War68]
    Warren, H.: Lower bounds for approximation of nonlinear manifolds. Trans. AMS 133, 167–178 (1968)zbMATHCrossRefMathSciNetGoogle Scholar
  39. [Weg87]
    Wegener, I.: The complexity of boolean functions. B. G. Teubner, and J. Wiley & Sons (1987)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Rafael Grimson
    • 1
    • 2
  1. 1.Dept. of MathematicsUniversity of Buenos Aires 
  2. 2.Dept. of Computer ScienceHasselt University 

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