Counting complexity classes for numeric computations. III: Complex projective sets

Foundations of Computational Mathematics

Volumn 5, Issue 4, 2005, Pages 351-387

  Bürgisser, Peter ^a   Cucker, Felipe ^b   Lotz, Martin ^a  

^a UNIVERSITY OF PADERBORN   (Germany)

^b CITY UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Complexity classes; Counting problems; Euler characteristic

Indexed keywords

COMPLEX NUMBER; COMPLEXITY CLASS; COUNTING COMPLEXITY; COUNTING PROBLEMS; EULER CHARACTERISTIC; NUMERIC COMPUTATION; SEMI-ALGEBRAIC SET; TURING MODEL;

COMPUTATIONAL METHODS;

COMPUTATIONAL COMPLEXITY;

EID: 31644451422     PISSN: 16153375     EISSN: 16153383     Source Type: Journal    
DOI: 10.1007/s10208-005-0146-x     Document Type: Conference Paper

References (36)

1. P. Aluffi, Computing characteristic classes of projective schemes, J. Symbolic Comput. 35(1) (2003), 3-19.
2. L. Blum, F. Cucker, M. Shub, and S. Smale, Algebraic settings for the problem "P ≠ NP?", in The Mathematics of Numerical Analysis, Lectures in Applied Mathematics, Vol. 32, pp. 125-144, Amer. Math. Soc., Providence, RI, 1996.
3. L. Blum, F. Cucker, M. Shub, and S. Smale, Complexity and Real Computation, Springer-Verlag, New York, 1998.
4. 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.
5. G. E. Bredon, Topology and Geometry, Graduate Texts in Mathematics, Vol. 139, Springer-Verlag, New York, 1997.
6. P. Bürgisser, Cook's versus Valiant's hypothesis, Theoret. Comput Sci. 235 (2000), 71-88.
7. P. Bürgisser, M. Clausen, and M. A. Shokrollahi, Algebraic Complexity Theory, Grundlehren der mathematischen Wissenschaften, Vol. 315, Springer-Verlag, Berlin, 1997.
8. P. Bürgisser and F. Cucker, Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets, in Proc. 36th Ann. ACM STOC, pp. 475-485, 2004. Full version at http://www.arxiv.org/abs/cs/cs.CC/0312007.
9. F. Cucker and D. Yu. Grigoriev, On the power of real Turing machines over binary inputs, SIAM J. Comput. 26 (1997), 243-254.
10. F. Cucker, M. Karpinski, P. Koiran, T. Lickteig, and K. Weither, On real Turing machines that toss coins, in Proc. 27th ACM STOC, Las Vegas, pp. 335-342, 1995.
11. F. Cucker and P. Koiran, Computing over the reals with addition and order: Higher complexity classes, J. Complexity. 11 (1995), 358-376.
12. A. Dickenstein, N. Fitchas, M. Giusti, and C. Sessa, The membership problem for unmixed polynomial ideals is solvable in single exponential time, Discrete Appl. Math. 33(1-3) (1991), 73-94.
13. A. Dimca, Singularities and Topology of Hypersurfaces, Universitext, Springer-Verlag, New York, 1992.
14. L. Fortnow, Counting complexity, in Complexity Theory Retrospective, II, pp. 81-107, Springer-Verlag, New York, 1997.
15. W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1993.
16. W. Fulton, Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 2, Springer-Verlag, Berlin, 1998.
17. M. Giusti and J. Heintz, La détermination des points isolés et de la dimension d'une varieté algébrique peut se faire en temps polynomial, in (D. Eisenbud and L. Robbiano, eds.), Proc. Cortona Conference on Computational Algebraic Geometry and Commutative Algebra, Cambridge University Press, Cambridge, 1993.
18. J. Harris, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 133, Springer-Verlag, New York, 1995.
19. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Springer-Verlag, New York, 1977.
20. A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.
21. O. H. Ibarra and S. Moran, Equivalence of straight-line programs, J. ACM 30 (1983), 217-228.
22. P. Koiran, Computing over the reals with addition and order, Theoret. Comput. Sci. 133 (1994), 35-47.
23. P. Koiran, Elimination of constants from machines over algebraically closed fields, J. Complexity, 13 (1997), 65-82.
24. P. Koiran, Randomized and deterministic algorithms for the dimension of algebraic varieties, in Proc. 38th FOCS, pp. 36-45, 1997.
25. P. Koiran, A weak version of the Blum, Shub and Smale model, J. Comput. System Sci. 54 (1997), 177-189.
26. P. Koiran, Elimination of parameters in the polynomial hierarchy, Theoret. Comput. Sci. 215 (1999), 289-304.
27. T. Krick and L. M. Pardo, Une approche informatique pour l'approximation Diophantienne, C. R. Acad. Sci. Paris 318 (1994), 407-412.
28. Y. N. Lakshman, A single exponential bound on the complexity of computing Gröbner bases of zero-dimensional ideals, in Effective Methods in Algebraic Geometry (Castiglioncello, 1990), Progress in Mathematics, Vol. 94, pp. 227-234, Birkhäuser Boston, Boston, MA, 1991.
29. A. Macintyre, Model completeness, in (J. Barwise, ed.), Handbook of Mathematical Logic, pp. 139-180, North-Holland, Amsterdam, 1977.
30. H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, 1986.
31. K. Meer, Counting problems over the reals, Theoret. Comput. Sci. 242 (2000), 41-58.
32. D. Mumford, Algebraic Geometry I: Complex Projective Varieties, Springer-Verlag, New York, 1976.
33. I. R. Shafarevich, Basic Algebraic Geometry, Springer-Verlag, New York, 1974.
34. L. G. Valiant, The complexity of computing the permanent, Theoret. Comput. Sci. 8 (1979), 189-201.
35. L. G. Valiant, The complexity of enumeration and reliability problems, SIAM J. Comput. 8 (1979), 410-421.
36. A. C. Yao, Algebraic decision trees and Euler characteristic, in Proc. 33rd FOCS, 1992.