Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets

Journal of Complexity

Volumn 22, Issue 2, 2006, Pages 147-191

  Bürgisser, Peter ^a   Cucker, Felipe ^b  

^a UNIVERSITY OF PADERBORN   (Germany)

^b CITY UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Betti numbers; Counting complexity; Euler characteristic; Geometric degree; Real complexity classes

Indexed keywords

COMPUTATIONAL COMPLEXITY; MATHEMATICAL MODELS; POLYNOMIALS; PROBLEM SOLVING; TOPOLOGY;

BETTI NUMBERS; COUNTING COMPLEXITY; REAL COMPLEXITY CLASSES;

SET THEORY;

EID: 31644440700     PISSN: 0885064X     EISSN: 10902708     Source Type: Journal    
DOI: 10.1016/j.jco.2005.11.001     Document Type: Article

References (77)

1. L. Babai, and S. Moran Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity classes J. Comput. System Sci. 36 2 1988 254 276
2. E. Bach Sheaf cohomology is # P -hard J. Symbolic Comput. 27 4 1999 429 433
3. J.L. Balcázar, J. Díaz, and J. Gabarró Structural Complexity I 1988 Springer Berlin
4. S. Basu On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets Discrete Comput. Geom. 22 1 1999 1 18
5. S. Basu, R. Pollack, and M.-F. Roy Computing roadmaps of semi-algebraic sets on a variety J. Amer. Math. Soc. 13 1999 55 82
6. S. Basu, R. Pollack, M.-F. Roy, Algorithms in Real Algebraic Geometry, Algorithms and Computation in Mathematics, vol. 10, Springer, Berlin, 2003.
7. R. Benedetti, and J.-J. Risler Real Algebraic and Semi-Algebraic Sets 1990 Berlin, Hermann Paris
8. L. Blum, F. Cucker, M. Shub, S. Smale, Algebraic settings for the problem " P ≠ NP ?", in: The Mathematics of Numerical Analysis, Lectures in Applied Mathematics, vol. 32, American Mathematical Society, Providence, RI, 1996, pp. 125-144.
9. L. Blum, F. Cucker, M. Shub, and S. Smale Complexity and Real Computation 1998 Springer Berlin
10. L. Blum, M. Shub, and S. Smale On a theory of computation and complexity over the real numbers Bull. Amer. Math. Soc. 21 1989 1 46
11. J. Bochnak, M. Coste, M.F. Roy, Géometrie algébrique réelle, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 12, Springer, Berlin, 1987.
12. A. Borel, and J.C. Moore Homology theory for locally compact spaces Michigan Math. J. 7 1960 137 159
13. G.E. Bredon, Topology and Geometry, GTM, vol. 139, Springer, Berlin, 1993.
14. J.W. Bruce Euler characteristics of real varieties Bull. London Math. Soc. 22 1990 547 552
15. P. Bürgisser, Completeness and Reduction in Algebraic Complexity Theory, Algorithms and Computation in Mathematics, vol. 7, Springer, Berlin, 2000.
16. P. Bürgisser Cook's versus Valiant's hypothesis Theoret. Comput. Sci. 235 2000 71 88
17. P. Bürgisser, M. Clausen, M.A. Shokrollahi, Algebraic Complexity Theory, Grundlehren der mathematischen Wissenschaften, vol. 315, Springer, Berlin, 1997.
18. P. Bürgisser, and F. Cucker Counting complexity classes for numeric computations I: semilinear sets SIAM J. Comput. 33 2003 227 260
19. P. Bürgisser, F. Cucker, M. Lotz, Counting complexity classes for numeric computations III: complex projective sets, Found. Comput. Math. 5(4) (2005) 351-387.
20. P. Bürgisser, M. Lotz, The complexity of computing the Hilbert polynomial of smooth equidimensional complex projective varieties, Found. Comput. Math., to appear.
21. J. Canny, Some algebraic and geometric computations in PSPACE, in: Proceedings of the 20th Annual ACM STOC, 1988, pp. 460-467.
22. F. Cucker On the complexity of quantifier elimination: the structural approach Comput. J. 36 1993 399 408
23. F. Cucker, and D.Yu. Grigoriev On the power of real Turing machines over binary inputs SIAM J. Comput. 26 1997 243 254
24. F. Cucker, M. Karpinski, P. Koiran, T. Lickteig, K. Werther, On real Turing machines that toss coins, in: Proceedings of the 27th ACM STOC, Las Vegas, 1995, pp. 335-342.
25. F. Cucker, and P. Koiran Computing over the reals with addition and order: higher complexity classes J. Complexity 11 1995 358 376
26. D. Eisenbud, and H. Levine An algebraic formula for the degree of a C ∞ map-germ Ann. Math. 106 1977 19 38
27. N. Fitchas, A. Galligo, and J. Morgenstern Precise sequential and parallel complexity bounds for quantifier elimination over algebraically closed fields J. Pure Appl. Algebra 67 1990 1 14
28. W. Fulton Introduction to Toric Varieties, Annals of Mathematical Studies 1993 Princeton University Press Princeton
29. F. Geerts Expressing the cone radius in the relational calculus with real polynomial constraints Discrete Comput. Geom. 30 4 2003 607 622
30. D.Yu. Grigoriev Complexity of deciding Tarski algebra J. Symbolic Comput. 5 1988 65 108
31. D.Yu. Grigoriev, and N. Vorobjov Solving systems of polynomial inequalities in subexponential time J. Symbolic Comput. 5 1988 37 64
32. D.Yu. Grigoriev, and N. Vorobjov Counting connected components of a semialgebraic set in subexponential time Comput. Complexity 2 2 1992 133 186
33. R. Hartshorne Algebraic Geometry 1977 GTM, Springer Berlin
34. A. Hatcher Algebraic Topology 2002 Cambridge University Press Cambridge
35. J. Heintz Definability and fast quantifier elimination in algebraically closed fields Theoret. Comput. Sci. 24 1983 239 277
36. J. Heintz, and J. Morgenstern On the intrinsic complexity of elimination theory J. Complexity 9 1993 471 498
37. J. Heintz, M.F. Roy, and P. Solernó Description of the connected components of a semialgebraic set in single exponential time Discrete Comput. Geom. 11 2 1994 121 140
38. J. Heintz, C.P. Schnorr, Testing polynomials which are hard to compute, in: Logic and Algorithmic: An International Symposium held in Honor of Ernst Specker, Monograph No. 30 de l'Enseign. Math., 1982, pp. 237-254.
39. H. Hironaka, Triangulation of algebraic sets, in: Algebraic Geometry (Proceedings of the Symposia in the Pure Mathematics vol. 29, Humboldt State University Arcata, California, 1974), Proceedings of the Symposia in Pure Mathematics, vol. 29, American Mathematial Society, 1975, pp. 165-185.
40. F. Hirzebruch New Topological Methods in Algebraic Geometry 1965 Springer Berlin
41. P. Koiran Computing over the reals with addition and order Theoret. Comput. Sci. 133 1994 35 47
42. P. Koiran, Hilbert's Nullstellensatz is in the polynomial hierarchy, J. Complexity 12 (1996) 273-286.
43. Additional information is contained in the DIMACS Technical Report 96-27, July 1996.
44. P. Koiran Elimination of constants from machines over algebraically closed fields J. Complexity 13 1997 65 82
45. P. Koiran, Randomized and deterministic algorithms for the dimension of algebraic varieties, in: Proceedings of the 38th FOCS, 1997, pp. 36-45.
46. P. Koiran A weak version of the Blum, Shub & Smale model J. Comput. System Sci. 54 1997 177 189
47. P. Koiran Elimination of parameters in the polynomial hierarchy Theoret. Comput. Sci. 215 1999 289 304
48. P. Koiran The real dimension problem is NP R -complete J. Complexity 15 2 1999 227 238
49. G.M. Kuper, J. Paredaens, and L. Libkin Constraint Databases 2000 Springer Berlin
50. I. Lakatos Proofs and Refutations: The Logic of Mathematical Discovery 1976 Cambridge University Press Cambridge
51. E.W. Mayr Some complexity results for polynomial ideals J. Complexity 13 1997 303 325
52. E.W. Mayr, and A.R. Meyer The complexity of the word problem for commutative semigroups and polynomial ideals Adv. Math. 46 1982 305 329
53. K. Meer Counting problems over the reals Theoret. Comput. Sci. 242 2000 41 58
54. J. Milnor, Morse theory, Annals of Mathematical Studies, vol. 51, Princeton University Press, Princeton, 1963.
55. J. Milnor, On the Betti numbers of real varieties. in: Proceedings of the AMS, vol. 15, 1964, pp. 275-280.
56. J. Milnor, Singular points of complex hypersurfaces, Annals of Mathematical Studies, vol. 61, Princeton University Press, Princeton, 1968.
57. D. Mumford Algebraic Geometry I: Complex Projective Varieties 1976 Springer Berlin
58. J.R. Munkres Elements of Algebraic Topology 1984 Addison-Wesley Publishing Company Menlo Park, CA
59. O.A. Oleǐnik Estimates of the Betti numbers of real algebraic hypersurfaces Math. Sb. (N.S.) 28 70 1951 635 640
60. O.A. Oleǐnik, and I.B. Petrovskii On the topology of real algebraic surfaces Izv. Akad. Nauk SSSR 13 1949 389 402
61. C.H. Papadimitriou Computational Complexity 1994 Addison-Wesley Reading, MA
62. E. Rannou The complexity of stratification computation Discrete Comput. Geom. 19 1998 47 78
63. J.H. Reif, Complexity of the mover's problem and generalizations, in: Proceedings of the 20th FOCS, 1979, pp. 421-427.
64. J.H. Reif Complexity of the generalized mover's problem J.T. Schwartz M. Sharir J. Hopcroft Planning, Geometry and Complexity of Robot Motion 1987 Ablex Publishing Corporation Norwood, NJ 267 281
65. J. Renegar On the computational complexity and geometry of the first-order theory of the reals, part I, II, III J. Symbolic Comput. 13 3 1992 255 352
66. I.R. Shafarevich Basic Algebraic Geometry 1974 Springer Berlin
67. E. Spanier Algebraic Topology 1966 MacGraw-Hill New York
68. N. Steenrod Topology of Fibre Bundles 1965 Princeton University Press Princeton
69. V. Strassen Die Berechnungskomplexität von elementarsymmetrischen Funktionen und von Interpolationskoeffizienten Numer. Math. 20 1973 238 251
70. Z. Szafraniec On the Euler characteristic of analytic and algebraic sets Topology 25 1986 411 414
71. R. Thom Sur l'homologie des variétés algébriques réelles S.S. Cairns Differential and Combinatorial Topology 1965 Princeton University Press Princeton 255 265
72. S. Toda PP is as hard as the polynomial-time hierarchy SIAM J. Comput. 21 2 1991 865 877
73. L.G. Valiant, Completeness classes in algebra, in: Proceedings of the 11th ACM STOC, 1979, pp. 249-261.
74. L.G. Valiant The complexity of computing the permanent Theoret. Comput. Sci. 8 1979 189 201
75. L.G. Valiant The complexity of enumeration and reliability problems SIAM J. Comput. 8 1979 410 421
76. L.G. Valiant, Reducibility by algebraic projections, in: Logic and Algorithmic: an International Symposium held in Honor of Ernst Specker, vol. 30, Monographs No. 30 de l'Enseign. Math. 1982, pp. 365-380.
77. A.C. Yao Algebraic decision trees and Euler characteristic Theoret. Comput. Sci. 141 1995 133 150