메뉴 건너뛰기




Volumn 110, Issue 2, 1996, Pages 131-184

Semi-algebraic decision complexity, the real spectrum, and degree

Author keywords

Decision trees; Degree method; Derivations; Real spectrum; Straight line programs

Indexed keywords


EID: 0030597930     PISSN: 00224049     EISSN: None     Source Type: Journal    
DOI: 10.1016/0022-4049(95)00105-0     Document Type: Article
Times cited : (4)

References (52)
  • 1
    • 85029969778 scopus 로고
    • manuscript, Universität Konstanz
    • [1] W. Baur, manuscript, Universität Konstanz (1989).
    • (1989)
    • Baur, W.1
  • 2
    • 0005478995 scopus 로고
    • The complexity of partial derivatives
    • [2] W. Baur and V. Strassen, The complexity of partial derivatives, Theoret. Comput. Sci. 22 (1983) 317-330.
    • (1983) Theoret. Comput. Sci. , vol.22 , pp. 317-330
    • Baur, W.1    Strassen, V.2
  • 3
    • 84968497743 scopus 로고
    • On the real spectrum of a ring and its applications to semi-algebraic geometry
    • [3] E. Becker, On the real spectrum of a ring and its applications to semi-algebraic geometry, Bull. Am. Math. Soc. (N. S.) 15 (1986) 19-60.
    • (1986) Bull. Am. Math. Soc. (N. S.) , vol.15 , pp. 19-60
    • Becker, E.1
  • 4
    • 0020879158 scopus 로고
    • Lower bounds for algebraic computation trees
    • Boston
    • [4] M. Ben-Or, Lower bounds for algebraic computation trees, Proc. 15th ACM STOC, Boston (1983) 80-86.
    • (1983) Proc. 15th ACM STOC , pp. 80-86
    • Ben-Or, M.1
  • 7
    • 0026979103 scopus 로고
    • Linear decision trees: Volume estimates and topological bounds
    • [7] A. Björner, L. Lovász and A. Yao, Linear decision trees: volume estimates and topological bounds, Proc. STOC'92 (1992) 170-177.
    • (1992) Proc. STOC'92 , pp. 170-177
    • Björner, A.1    Lovász, L.2    Yao, A.3
  • 8
    • 0039032262 scopus 로고
    • Géométrie algébrique réelle, ergebnisse der mathematik und ihrer grenzgebiete, 3
    • Springer, Berlin
    • [8] J. Bochnak, M. Coste and M.-F. Roy, Géométrie algébrique réelle, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 12 (Springer, Berlin, 1987).
    • (1987) Folge, Band , vol.12
    • Bochnak, J.1    Coste, M.2    Roy, M.-F.3
  • 9
    • 84968516134 scopus 로고
    • On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines
    • [9] L. Blum, M. Shub and S. Smale, On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines, Bull. Amer. Math. Soc. 21 (1989) 1-46.
    • (1989) Bull. Amer. Math. Soc. , vol.21 , pp. 1-46
    • Blum, L.1    Shub, M.2    Smale, S.3
  • 10
    • 0000147250 scopus 로고
    • On randomized semi-algebraic decision complexity
    • [10] P. Bürgisser, M. Karpinski and T. Lickteig, On randomized semi-algebraic decision complexity, J. Complexity 9 (1993) 231-251.
    • (1993) J. Complexity , vol.9 , pp. 231-251
    • Bürgisser, P.1    Karpinski, M.2    Lickteig, T.3
  • 11
    • 38249009716 scopus 로고
    • Verification complexity of linear prime ideals
    • [11] P. Bürgisser and T. Lickteig, Verification complexity of linear prime ideals, J. Pure Appl. Algebra 81 (1992) 247-267.
    • (1992) J. Pure Appl. Algebra , vol.81 , pp. 247-267
    • Bürgisser, P.1    Lickteig, T.2
  • 12
    • 38249009027 scopus 로고
    • Test complexity of generic polynomials
    • [12] P. Bürgisser, T. Lickteig and M. Shub, Test complexity of generic polynomials, J. Complexity 8 (1992) 203-215.
    • (1992) J. Complexity , vol.8 , pp. 203-215
    • Bürgisser, P.1    Lickteig, T.2    Shub, M.3
  • 14
    • 0011674002 scopus 로고
    • Introduction to algebraic geometry and algebraic groups
    • North Holland, Amsterdam
    • [14] M. Demazure and P. Gabriel, Introduction to algebraic geometry and algebraic groups, Math. Stud. (North Holland, Amsterdam, 1980).
    • (1980) Math. Stud.
    • Demazure, M.1    Gabriel, P.2
  • 16
    • 85029972465 scopus 로고    scopus 로고
    • Lower bounds on testing membership to a polyhedron by algebraic decision trees
    • to appear
    • [16] D. Grigoriev, M. Karpinski and N. Vorobjov, Lower bounds on testing membership to a polyhedron by algebraic decision trees, Proc. STOC'94, to appear.
    • Proc. STOC'94
    • Grigoriev, D.1    Karpinski, M.2    Vorobjov, N.3
  • 18
    • 0000627865 scopus 로고
    • Definability and fast quantifier elimination over algebraically closed fields
    • [18] J. Heintz, Definability and fast quantifier elimination over algebraically closed fields, Theoret. Comput. Sci. 24 (1983) 239-278.
    • (1983) Theoret. Comput. Sci. , vol.24 , pp. 239-278
    • Heintz, J.1
  • 19
    • 0011538773 scopus 로고
    • Lower bounds for the non-linear complexity of algebraic computation trees with integer inputs
    • [19] M.D. Hirsch, Lower bounds for the non-linear complexity of algebraic computation trees with integer inputs, Comput. Complexity 1 (1991) 257-268.
    • (1991) Comput. Complexity , vol.1 , pp. 257-268
    • Hirsch, M.D.1
  • 20
    • 85029968393 scopus 로고
    • Tech. Report No. 10/93, Sonderforschungsbereich 124, Saarbrücken-Kaiserslautern, Germany
    • [20] G. Hotz and J. Sellen, On algebraic computation trees and Betti numbers, Tech. Report No. 10/93, Sonderforschungsbereich 124, Saarbrücken-Kaiserslautern, Germany (1993).
    • (1993) On Algebraic Computation Trees and Betti Numbers
    • Hotz, G.1    Sellen, J.2
  • 22
    • 0003245465 scopus 로고
    • Einführung in die reelle algebra
    • Aufbaukurs Mathematik, Vieweg
    • [22] M. Knebusch and C. Scheiderer, Einführung in die reelle Algebra, Vieweg-Studium 63 (Aufbaukurs Mathematik, Vieweg, 1989).
    • (1989) Vieweg-Studium , vol.63
    • Knebusch, M.1    Scheiderer, C.2
  • 23
    • 0009979695 scopus 로고
    • Einführung in die kommutative algebra und algebraische geometrie
    • Aufbaukurs Mathematik, Vieweg
    • [23] E. Kunz, Einführung in die kommutative Algebra und algebraische Geometrie, Vieweg-Studium 46 (Aufbaukurs Mathematik, Vieweg, 1980).
    • (1980) Vieweg-Studium , vol.46
    • Kunz, E.1
  • 24
    • 0004270841 scopus 로고
    • Tech. Report TR-90-052 Int. Comp. Science Inst., Berkeley and Univ. Tübingen, Habilitationsschrift
    • [24] T. Lickteig, On semialgebraic decision complexity, Tech. Report TR-90-052 Int. Comp. Science Inst., Berkeley (1990) and Univ. Tübingen, Habilitationsschrift.
    • (1990) On Semialgebraic Decision Complexity
    • Lickteig, T.1
  • 26
    • 0017254690 scopus 로고
    • Taylor expansion of the accumulated rounding error
    • [26] S. Linnainmaa, Taylor expansion of the accumulated rounding error, BIT 16 (1976) 146-160.
    • (1976) BIT , vol.16 , pp. 146-160
    • Linnainmaa, S.1
  • 28
    • 0021462945 scopus 로고
    • A polynomial linear search algorithm for the n-dimensional knapsack problem
    • [28] F. Meyer auf der Heide, A polynomial linear search algorithm for the n-dimensional knapsack problem, J. ACM 31 (1984) 668-676.
    • (1984) J. ACM , vol.31 , pp. 668-676
    • Meyer Auf Der Heide, F.1
  • 29
    • 84966260816 scopus 로고
    • On the Betti numbers of real varieties
    • [29] J. Milnor, On the Betti numbers of real varieties, Proc. Amer. Math. Soc. 15 (1964) 275-280.
    • (1964) Proc. Amer. Math. Soc. , vol.15 , pp. 275-280
    • Milnor, J.1
  • 30
    • 0001933332 scopus 로고
    • Lower bounds for arithmetic networks II: Sum of Betti numbers
    • to appear
    • [30] J.L. Montaǹa, J.E. Morais and L.M. Pardo, Lower bounds for arithmetic networks II: sum of Betti numbers, AAECC 7 (1995), to appear.
    • (1995) AAECC , vol.7
    • Montaǹa, J.L.1    Morais, J.E.2    Pardo, L.M.3
  • 31
    • 0002392671 scopus 로고
    • Lower bounds for arithmetic networks
    • [31] J.L. Montaña and L.M. Pardo, Lower bounds for arithmetic networks, AAECC 4 (1993) 1-24.
    • (1993) AAECC , vol.4 , pp. 1-24
    • Montaña, J.L.1    Pardo, L.M.2
  • 34
    • 0001895223 scopus 로고
    • How to compute fast a function and all its derivatives
    • [34] J. Morgenstern, How to compute fast a function and all its derivatives, SIGACT News 16 (1985) 60-62.
    • (1985) SIGACT News , vol.16 , pp. 60-62
    • Morgenstern, J.1
  • 36
    • 0003197929 scopus 로고
    • Lectures on formally real fields
    • Springer, Berlin
    • [36] A. Prestel, Lectures on formally real fields, Lecture Notes in Mathematics, Vol. 1093 (Springer, Berlin, 1984).
    • (1984) Lecture Notes in Mathematics , vol.1093
    • Prestel, A.1
  • 37
    • 0003438092 scopus 로고
    • Einführung in die mathematische logik und modelltheorie
    • Aufbaukurs Mathematik, Vieweg
    • [37] A. Prestel, Einführung in die Mathematische Logik und Modelltheorie, Vieweg-Studium 60 (Aufbaukurs Mathematik, Vieweg, 1986).
    • (1986) Vieweg-Studium , vol.60
    • Prestel, A.1
  • 38
    • 0000812271 scopus 로고
    • Faisceau structural sur le spectre réel et fonctions de Nash
    • Géométrie algébrique réelle et formes quadratiques, Springer, Berlin
    • [38] M.-F. Roy, Faisceau structural sur le spectre réel et fonctions de Nash, in: Géométrie algébrique réelle et formes quadratiques, Lecture Notes in Mathematics, Vol. 959 (Springer, Berlin, 1982) 406-432.
    • (1982) Lecture Notes in Mathematics , vol.959 , pp. 406-432
    • Roy, M.-F.1
  • 39
    • 0011668791 scopus 로고
    • An extension of Strassen's degree bound
    • [39] C.P. Schnorr, An extension of Strassen's degree bound, SIAM J. Comput. 10 (1981) 371-382.
    • (1981) SIAM J. Comput. , vol.10 , pp. 371-382
    • Schnorr, C.P.1
  • 41
    • 0011609133 scopus 로고
    • An elementary proof for Strassen's degree bound
    • [41] A. Schönhage, An elementary proof for Strassen's degree bound, Theoret. Comput. Sci. 3 (1976) 267-272.
    • (1976) Theoret. Comput. Sci. , vol.3 , pp. 267-272
    • Schönhage, A.1
  • 43
    • 0002943521 scopus 로고
    • Lower bound for algebraic decision trees
    • [43] J.M. Steele and A.C.C. Yao, Lower bound for algebraic decision trees, J. Algorithms 3 (1982) 1-8.
    • (1982) J. Algorithms , vol.3 , pp. 1-8
    • Steele, J.M.1    Yao, A.C.C.2
  • 44
    • 0022328333 scopus 로고
    • The complexity of evaluating interpolation polynomials
    • [44] H.J. Stoß, The complexity of evaluating interpolation polynomials, Theoret. Comput. Sci. 41 (1985) 319-323.
    • (1985) Theoret. Comput. Sci. , vol.41 , pp. 319-323
    • Stoß, H.J.1
  • 45
    • 0015491022 scopus 로고
    • Berechnung und programm I
    • [45] V. Strassen, Berechnung und Programm I, Acta Inform, 1 (1973) 320-335.
    • (1973) Acta Inform. , vol.1 , pp. 320-335
    • Strassen, V.1
  • 46
    • 0001063303 scopus 로고
    • Die berechnungskomplexität von elementarsymmetrischen funktionen und interpolations-koeffizienten
    • [46] V. Strassen, Die Berechnungskomplexität von elementarsymmetrischen Funktionen und Interpolations-koeffizienten, Num. Math. 20 (1973) 238-251.
    • (1973) Num. Math. , vol.20 , pp. 238-251
    • Strassen, V.1
  • 47
    • 0001977576 scopus 로고
    • The computational complexity of continued fractions
    • [47] V. Strassen, The computational complexity of continued fractions, SIAM J. Comput. 12 (1983) 1-27.
    • (1983) SIAM J. Comput. , vol.12 , pp. 1-27
    • Strassen, V.1
  • 48
    • 0002586126 scopus 로고
    • Sur l'homologie des variétés algébriques réelles
    • S.S. Cairns, ed., Princton Univ. Press, Princeton
    • [48] R. Thom, Sur l'homologie des variétés algébriques réelles, in: S.S. Cairns, ed., Differential and Combinatorial Topology (Princton Univ. Press, Princeton, 1965) 255-265.
    • (1965) Differential and Combinatorial Topology , pp. 255-265
    • Thom, R.1
  • 49
    • 0003539886 scopus 로고
    • Lecture Notes, TATA Institute of Fundamental Research, Bombay, No. 74
    • [49] W. Vogel, On Results on Bézout's Theorem, Lecture Notes, TATA Institute of Fundamental Research, Bombay, No. 74 (1984).
    • (1984) On Results on Bézout's Theorem
    • Vogel, W.1
  • 50
    • 0026202856 scopus 로고
    • Lower bounds for algebraic computation trees with integer inputs
    • [50] A.C.C. Yao, Lower bounds for algebraic computation trees with integer inputs, SIAM J. Comput. 20 (1989) 655-668.
    • (1989) SIAM J. Comput. , vol.20 , pp. 655-668
    • Yao, A.C.C.1
  • 51
    • 85065717612 scopus 로고
    • Algebraic decision trees and Euler characteristics
    • [51] A.C.C. Yao, Algebraic decision trees and Euler characteristics, Proc. FOCS'92 (1992) 268-277.
    • (1992) Proc. FOCS'92 , pp. 268-277
    • Yao, A.C.C.1
  • 52
    • 3242844427 scopus 로고
    • Decision trees and Betti numbers
    • to appear
    • [52] A.C.C. Yao, Decision trees and Betti numbers, Proc. STOC'94 (1994), to appear.
    • (1994) Proc. STOC'94
    • Yao, A.C.C.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.