On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets

Discrete and Computational Geometry

Volumn 22, Issue 1, 1999, Pages 1-18

  Basu, S ^a  

^a UNIVERSITY OF MICHIGAN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033436565     PISSN: 01795376     EISSN: None     Source Type: Journal    
DOI: 10.1007/PL00009443     Document Type: Article

References (32)

1. A. I. Barvinok, On the Betti numbers of semi-algebraic sets defined by few quadratic inequalities, Mathematische Zeitschrift, 225 (1997), 231-244.
2. S. Basu, On bounding the Betti numbers and computing the Euler characteristics of semi-algebraic sets, Proc. 28th Annual ACM Symposium on the Theory of Computing, pp. 408-417, (1996).
3. S. Basu, R. Pollack, and M.-F. Roy, On the combinatorial and algebraic complexity of quantifier elimination, Journal of the ACM, 43(6) (1996), 1002-1045.
4. S. Basu, R. Pollack, and M.-F. Roy, On the number of cells defined by a family of polynomials on a variety, Mathematika, 43 (1996), 120-126.
5. S. Basu, R. Pollack, and M.-F. Roy, Computing roadmaps of semi-algebraic sets on a variety (extended abstract), Proc. First Conference on the Foundations of Computational Mathematics, F. Cucker and M. Shub, eds., Springer-Verlag, New York, 1997, pp. 1-15 (full version to appear in the Journal of the AMS).
6. J. Bochnak, M. Coste, and M.-F. Roy, Géométrie algébrique réelle. Springer-Verlag, New York, 1987.
7. J. Canny, Computing road maps in general semi-algebraic sets, Computer Journal 36 (1993), 504-514.
8. J. Canny, D. Grigoriev, and N. Vorobjov, Finding connected components of a semi-algebraic set in sub-exponential time, Applicable Algebra in Engineering, Communication and Computing, 2 (1992), 217-238.
9. G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Lecture Notes in Computer Science, vol. 33, pp. 515-532, Springer-Verlag, Berlin, 1975.
10. S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton University Press, Princeton, NJ, 1952.
11. A. Gabrielov, Multiplicities of zeros of polynomials on trajectories of polynomial vector fields and bounds on degree of nonholonomy, Mathematical Research Letters, 2(4) (1995), 437-451.
12. d, Discrete and Computational Geometry, 1 (1986), 219-227.
13. J. E. Goodman, R. Pollack, and R. Wenger, Bounding the number of geometric permutations induced by k-transversals, Journal of Combinatorial Theory, Series A, 75 (1996), 187-197.
14. L. Gournay and J. J. Risler, Construction of roadmaps of semi-algebraic sets, Technical Report, DMI, Ecole Normale Superieure, Paris, (1992).
15. D. Grigoriev, N. Vorobjov, and J. Canny, Counting connected components of a semi-algebraic set in sub-exponential time, Computational Complexity, 2(2) (1992), 133-186.
16. R. M. Hardt, Semi-algebraic local triviality in semi-algebraic mappings, American Journal of Mathematics, 102 (1980), 291-302.
17. J. Heintz, T. Recio, and M.-F. Roy, Algorithms in real algebraic geometry and applications to computational geometry Discrete and Computational Geometry: Selected Papers from the DIMACS Special Year, J. E. Goodman, R. Pollack, and W. Steiger, eds. AMS, Providence, RI, and ACM, New York, 1991, vol. 6, pp. 137-164.
18. J. Heintz, M.-F. Roy, and P. Solernó, Single exponential path finding in semi-algebraic sets, II: The general case, in Algebraic Geometry and Its Applications, C. L. Bajaj, ed., Springer-Verlag, New York, 1993, pp. 467-481.
19. J. Heintz, M.-F. Roy, and P. Solernó, Description of the connected components of a semi-algebraic set in single exponential time, Discrete and Computational Geometry, 11 (1994), 121-140.
20. A. G. Khovansky, Fewnomials, Translations of Mathematical Monographs, Vol. 88, AMS, Providence, RI, 1991.
21. M. Knebusch, Semi-algebraic topology in the last ten years, in Real Algebraic Geometry, M. Coste, L. Mahe, and M.-F. Roy, eds., Lecture Notes in Mathematics, vol. 1524, pp. 1-36, Springer-Verlag, Berlin, 1992.
22. J. Milnor, Morse Theory, Annals of Mathematical Studies, Princeton University Press, Princeton, NJ, 1963.
23. J. Milnor, On the Betti numbers of real varieties, Proceedings of the AMS 15 (1964), 275-280.
24. J. L. Montana, J. E. Morais, and L. M. Pardo, Lower bounds for arithmetic networks II: Sum of Betti numbers, Applicable Algebra in Engineering, Communication and Computing, 7 (1) (1996), 41-51.
25. O. A. Oleinik and I. B. Petrovsky, On the topology of real algebraic surfaces, Izvestiya Akademii Nauk SSSR 13 (1949), 389-402.
26. P. Pedersen M.-F. Roy, and A. Szpirglas, Counting real zeros in the multivariate case, in Computanonal Algebraic Geometry, F. Eyssette and A. Galligo, eds. Progress in Mathematics, vol. 109, pp. 203-224, Birkhauser, Basel, 1993.
27. J. Schwartz and M. Sharir, On the "piano movers" problem, II. General techniques for computing topological properties of real algebraic manifolds, Advances in Applied Mathematics, 4 (1983), 298-351.
28. E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
29. R. Thorn, Sur l'omologie des varietes algebriques reelles, in Differential and Combinatonal Topology, S. S. Cairns, ed., Princeton University Press, Princeton, NJ, 1965, pp. 255-265.
30. H. Warren, Lower bounds for approximation by nonlinear manifolds, Transactions of the AMS, 133 (1968), 167-178.
31. A. C. Yao, Decision tree complexity and Betti numbers, Proc. 26th ACM Symposium on Theory of Computing, pp. 615-624, 1994.
32. T. Zell, Betti numbers of semi-Pfaffian sets, Preprint.