Artículo

Estamos trabajando para incorporar este artículo al repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

Abstract:

We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of real projective varieties. Here numerical means that the algorithm is numerically stable (in a sense to be made precise). Its cost depends on the condition of the input as well as on its size and is singly exponential in the number of variables (the dimension of the ambient space) and polynomial in the condition and the degrees of the defining polynomials. In addition, we show that outside of an exceptional set of measure exponentially small in the size of the data, the algorithm takes exponential time. © 2017, SFoCM.

Registro:

Documento: Artículo
Título:Computing the Homology of Real Projective Sets
Autor:Cucker, F.; Krick, T.; Shub, M.
Filiación:Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong
Departamento de Matemática & IMAS, Univ. de Buenos Aires & CONICET, Buenos Aires, Argentina
Department of Mathematics, City College and the Graduate Center of CUNY, New York, NY, United States
Palabras clave:Complexity; Condition; Exponential time; Homology groups; Real projective varieties; Computational methods; Complexity; Condition; Exponential time; Homology groups; Real projective varieties; Mathematical techniques
Año:2018
Volumen:18
Número:4
Página de inicio:929
Página de fin:970
DOI: http://dx.doi.org/10.1007/s10208-017-9358-8
Título revista:Foundations of Computational Mathematics
Título revista abreviado:Found. Comput. Math.
ISSN:16153375
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_16153375_v18_n4_p929_Cucker

Referencias:

  • Allgower, E.L., Georg, K., (1990) Numerical Continuation Methods, , Springer-Verlag
  • Amelunxen, D., Lotz, M., Average-case complexity without the black swans (2016) To Appear at J. Compl
  • Basu, S., Computing the top Betti numbers of semialgebraic sets defined by quadratic inequalities in polynomial time (2008) Found. Comput. Math., 8 (1), pp. 45-80
  • Basu, S., Pollack, R., Roy, M.-F., Computing the first Betti number of a semi-algebraic set (2008) Found. Comput. Math., 8 (1), pp. 97-136
  • Björner, A., Topological methods (1995) Handbook of Combinatorics, pp. 1819-1872. , Graham R, Grotschel M, Lovasz L, (eds), North-Holland, Amsterdam
  • Blum, L., Cucker, F., Shub, M., Smale, S., (1998) Complexity and Real Computation, , Springer-Verlag
  • Blum, L., Shub, M., Smale, S., On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines (1989) Bulletin of the Amer. Math. Soc., 21, pp. 1-46
  • Bürgisser, P., Cucker, F., Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets (2006) J. Compl., 22, pp. 147-191
  • Bürgisser, P., Cucker, F., Exotic quantifiers, complexity classes, and complete problems (2009) Found. Comput. Math., 9, pp. 135-170
  • Bürgisser, P., Cucker, F., (2013) Condition, 349. , Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin
  • Cheung, D., Cucker, F., Solving linear programs with finite precision: II (2006) Algorithms. J. Compl., 22, pp. 305-335
  • Collins, G.E., (1975) Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Deccomposition, 33, pp. 134-183. , of Lect. Notes in Comp. Sci., Springer-Verlag
  • Cucker, F., Approximate zeros and condition numbers (1999) J. Compl., 15, pp. 214-226
  • Cucker, F., Diao, H., Wei, Y., Smoothed analysis of some condition numbers (2006) Numer. Lin. Alg. Appl., 13, pp. 71-84
  • Cucker, F., Krick, T., Malajovich, G., Wschebor, M., A numerical algorithm for zero counting. I (2008) J. Compl., 24, pp. 582-605
  • Cucker, F., Krick, T., Malajovich, G., Wschebor, M., A numerical algorithm for zero counting. II (2009) J. Fixed Point Theory Appl., 6, pp. 285-294
  • Cucker, F., Krick, T., Malajovich, G., Wschebor, M., A numerical algorithm for zero counting. III (2012) Adv. Applied Math., 48, pp. 215-248
  • Cucker, F., Peña, J., A primal-dual algorithm for solving polyhedral conic systems with a finite-precision machine (2002) SIAM J. Optim., 12, pp. 522-554
  • Cucker, F., Smale, S., Complexity estimates depending on condition and round-off error (1999) Journal of the ACM, 46, pp. 113-184
  • D’Andrea, C., Krick, T., Sombra, M., Heights of varieties in multiprojective spaces and arithmetic Nullstellensätze (2013) Ann. Sci. Éc. Norm. Supér. (4), 46 (4), pp. 549-627. , 2013
  • Demmel, J., The probability that a numerical analysis problem is difficult (1988) Math. Comp., 50, pp. 449-480
  • Edelsbrunner, H., Harer, J.L., Computational topology. American Mathematical Society, Providence, RI, 2010 An Introduction.
  • Kostlan, E., Complexity theory of numerical linear algebra (1988) J. of Computational and Applied Mathematics, 22, pp. 219-230
  • Lotz, M., On the volume of tubular neighborhoods of real algebraic varieties (2015) Proc. Amer. Math. Soc., 143 (5), pp. 1875-1889
  • Niyogi, P., Smale, S., Weinberger, S., Finding the homology of submanifolds with high confidence from random samples (2008) Discrete Comput. Geom., 39, pp. 419-441
  • Noferini, V., Townsend, A., Numerical instability of resultant methods for multidimensional rootfinding To Appear at SIAM J. Num. Analysis
  • Renegar, J., On the computational complexity and geometry of the first-order theory of the reals. Part I (1992) Journal of Symbolic Computation, 13, pp. 255-299
  • Renegar, J., Some perturbation theory for linear programming (1994) Math. Program., 65, pp. 73-91
  • Renegar, J., Incorporating condition measures into the complexity theory of linear programming (1995) SIAM J. Optim., 5, pp. 506-524
  • Renegar, J., Linear programming, complexity theory and elementary functional analysis (1995) Math. Program., 70, pp. 279-351
  • Scheiblechner, P., On the complexity of deciding connectedness and computing Betti numbers of a complex algebraic variety (2007) J. Complexity, 23 (3), pp. 359-379
  • Scheiblechner, P., Castelnuovo-Mumford regularity and computing the de Rham cohomology of smooth projective varieties (2012) Found. Comput. Math., 12 (5), pp. 541-571
  • Shub, M., Smale, S., Complexity of Bézout’s Theorem I: geometric aspects (1993) Journal of the Amer. Math. Soc., 6, pp. 459-501
  • Shub, M., Smale, S., Complexity of Bézout’s Theorem II: Volumes and probabilities (1993) Computational Algebraic Geometry, 109, pp. 267-285. , F. Eyssette, A. Galligo, Birkhäuser
  • Shub, M., Smale, S., Complexity of Bézout’s Theorem III: condition number and packing (1993) Journal of Complexity, 9, pp. 4-14
  • Shub, M., Smale, S., Complexity of Bézout’s Theorem V: polynomial time (1994) Theoret. Comp. Sci., 133, pp. 141-164
  • Shub, M., Smale, S., Complexity of Bézout’s Theorem IV: probability of success; extensions (1996) SIAM J. of Numer. Anal., 33, pp. 128-148
  • Smale, S., Newton’s method estimates from data at one point (1986) The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics, , R. Ewing, K. Gross, C. Martin, editors, Springer-Verlag
  • Storjohann, A., Nearly optimal algorithms for computing Smith normal forms of integer matrices (1996) Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC’96), pp. 267-274. , ACM Press
  • Wüthrich, H.R., Ein Entscheidungsverfahren für die Theorie der reell-abgeschlossenen Körper (1976) E. Specker and V. Strassen, Editors, Komplexität Von Entscheidungsproblemen, Volume 43 of Lect. Notes in Comp. Sci, pp. 138-162. , Springer-Verlag

Citas:

---------- APA ----------
Cucker, F., Krick, T. & Shub, M. (2018) . Computing the Homology of Real Projective Sets. Foundations of Computational Mathematics, 18(4), 929-970.
http://dx.doi.org/10.1007/s10208-017-9358-8
---------- CHICAGO ----------
Cucker, F., Krick, T., Shub, M. "Computing the Homology of Real Projective Sets" . Foundations of Computational Mathematics 18, no. 4 (2018) : 929-970.
http://dx.doi.org/10.1007/s10208-017-9358-8
---------- MLA ----------
Cucker, F., Krick, T., Shub, M. "Computing the Homology of Real Projective Sets" . Foundations of Computational Mathematics, vol. 18, no. 4, 2018, pp. 929-970.
http://dx.doi.org/10.1007/s10208-017-9358-8
---------- VANCOUVER ----------
Cucker, F., Krick, T., Shub, M. Computing the Homology of Real Projective Sets. Found. Comput. Math. 2018;18(4):929-970.
http://dx.doi.org/10.1007/s10208-017-9358-8