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International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes

AAECC 1995: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes pp 33–69Cite as

How lower and upper complexity bounds meet in elimination theory

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References

  1. J.L. Balcazar, J.L. Díaz and J. Gabarró. “Structural Complexity I”. EATCS Mon. on Theor. Comp. Sci. 11, Springer (1988).

    Google Scholar 

  2. W. Baur, V. Strassen. “The complexity of partial derivatives”. Theoret. Comput. Sci. 22 (1982) 317–330.

    Google Scholar 

  3. D. Bayer. “The Division Algorithm and the Hilbert Scheme”. Ph. D. Thesis. Harvard University (1982)

    Google Scholar 

  4. D. Bayer, D. Mumford. “What can be computed in algebraic geometry?”. In Computational Algebraic Geometry and Commutative Algebra”, Proc. of the Cortona Conf. on Comp. Algebraic Geometry and Commutative Algebra, D. Eisenbud and L. Robbiano (eds.). Symposia Matematica, vol. XXXIV, Istituto Nazionale di Alta Matematica, Cambridge University Press (1993) 1–49.

    Google Scholar 

  5. E. Becker, J.P. Cardinal, M.F. Roy, Z. Szafraniec. “Multivariate Bezoutians, Kronecker symbol and Eisenbud-Levine formula”. To appear in Proc. MEGA'94, Brikhaüsser Progress in Math.

    Google Scholar 

  6. M. Ben-Or. “Lower Bounds for Algebraic Computation Trees.” In ACM 15 th Symposium on Theory of Computation, (1983) 80–86.

    Google Scholar 

  7. S.J. Berkowitz.“On computing the determinant in small parallel time using a small number of processors”, Inf. Proc. Letters 18 (1984) 147–150.

    Google Scholar 

  8. C. Berenstein and A. Yger. “Effective Bézout identities in Q[X 1,...,X n]”, Acta Math., 166 (1991) 69–120.

    Google Scholar 

  9. C. Berenstein and A. Yger. “Une Formule de Jacobi et ses conséquences”, Ann. Sci. E.N.S., 4 ieme série, 24 (1991) 363–377.

    Google Scholar 

  10. L. J. Billera, I. M. Gel'fand and B. Sturmfels. “Duality and Minors of Secondary Polyhedra.” J. Combinatorial Theory, Vol. 57, No. 2 (1993) 258–268.

    Google Scholar 

  11. 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”. Bulletin of the Amer. Math. Soc., vol.21, n. 1, (1989) 1–46.

    Google Scholar 

  12. E. Borel. “La définition en mathématiques”. In Les Grands Courants de la Pensée Mathématique, Cahiers du Sud, Paris, (1948), 24–34.

    Google Scholar 

  13. A. Borodin. “On Relating Time and Space to Size and Depth”. SIAM J. on Comp. 6 (1977) 733–744.

    Google Scholar 

  14. A. Borodin, J. Munro. “The Computational Complexity of Algebraic and Numeric Problems”. Elsevier, NY, (1972).

    Google Scholar 

  15. J.-B. Bost, H. Gillet and C. Soulé. “Un analogue arithmétique du théorème de Bézout”, C.R. Acad. Sci. Paris, t. 312, Série I (1991) 845–848.

    Google Scholar 

  16. J.-B. Bost, H. Gillet and C. Soulé. “Heights of projective varieties and positive Green forms”, Manuscrit I.H.E.S. (1993).

    Google Scholar 

  17. D. W. Brownawell. “Bounds for the degree in the Nullstellensatz”, Annals of Math. 126 (1987) 577–591.

    Google Scholar 

  18. D.W. Brownawell. “Local Diophantine Nullstellen inequalities”, J. Amer. Math. Soc. 1 (1988) 311–322.

    Google Scholar 

  19. B. Buchberger. “Gröbner bases: An algorithmic method in polynomial ideal theory”. In Multidimensional System Theory (N.K. Bose, ed.), Reidel, Dordrecht (1985) 374–383.

    Google Scholar 

  20. L. Caniglia, A. Galligo and J. Heintz. “Borne simplement exponentielle pour les degrés dans le théorème des zéros sur un corps de caractéristique quelconque”, C.R. Acad. Sci. Paris, t. 307 Série I (1988) 255–258.

    Google Scholar 

  21. L. Caniglia, A. Galligo and J. Heintz. “Some new effectivity bounds in computational geometry”. In Proc. AAECC-6, (T. Mora. ed.), Springer LNCS 357 (1989) 131–151.

    Google Scholar 

  22. J.P. Cardinal. “Dualité et algorithmes itératives pour la solution des systémes polynomiaux”. Thése, U. de Rennes I (1993).

    Google Scholar 

  23. F. Cucker, J.L. Montaña, L.M. Pardo.“Time Bounded Computations over the reals”. Int. J. of Alg. and Comp., Vol.2, No. 4, (1992) 395–408.

    Google Scholar 

  24. F. Cucker, M. Shub, S. Smale. “Separation of complexity lasses in Koiran's weak model”. Theor. Comp. Sci. 133, N. 1, (1994) 3–15.

    Google Scholar 

  25. J. Davenport, J. Heintz. “Real quantifier elimination is doubly exponential”. J. Symbolic Computation 5 (1988) 29–35. Reprinted in: Algorithms in Real Algebraic Geometry, ed. by Denis A. Arnon and Bruno Buchberger, Academic Press (1988) 29–35.

    Google Scholar 

  26. D. Duval.“Diverses Questions Relatives au Cacul Formel avec des Nombres Algèbriques”. Thèse d' Etat, Grenoble (1987).

    Google Scholar 

  27. N. Fitchas, M. Giusti and F. Smietanski. “Sur la complexité du théorème des zéros”, Manuscript Ecole Polytechnique (1993).

    Google Scholar 

  28. N. Fitchas, A. Galligo, J. Morgenstern. “Algorithmes rapides en séquentiel et en parallele pour l'élimination de quantificateurs en géométrie élémentaire”. Seminaire de Structures Algebriques Ordennes, Université de Paris VII (1987).

    Google Scholar 

  29. M. Garey, D. Johnson. “Computers and Intractability: A Guide to the Theory of NP-completeness”. Freeman, San Francisco (1979).

    Google Scholar 

  30. J. von zur Gathen. “Parallel algorithms for algebraic problems”. SIAM J. Comput., Vol. 13, No. 4 (1984) 802–824

    Google Scholar 

  31. J.Von zur Gathen.“Parallel arithmetic computations: a survey”. In Proc. 13 th Mathematical Foundations on computer Science, (1986) 93–112.

    Google Scholar 

  32. J. von zur Gathen. “Algebraic Complexity Theory”. Annual Review of Computer Science, vol. 3 (1988).

    Google Scholar 

  33. A.O. Gel'fond. “Transcendental and Algebraic Numbers”. Dover, NY (1960).

    Google Scholar 

  34. P. Gianni, T. Mora. “Algebraic solution of systems of polynomial equations using Gröbner bases”. In Proc. AAECC-5, LNCS 356 (1989) 247–257.

    Google Scholar 

  35. M. Giusti and J. Heintz. “Algorithmes-disons rapides-pour la décomposition d'une variété algébrique en composantes irréductibles et équidimensionelles”, In Proc. MEGA '90, Birkhäuser Progress in Math. 94, (T. Mora and C. Traverso, eds.) (1991) 169–193.

    Google Scholar 

  36. M.Giusti and J.Heintz. “La détermination des points isolés et de la dimension d'une variété algébrique peut se faire en temps polynomial”. In Computational Algebraic Geometry and Commutative Algebra”, Proc. of the Cortona Conf. on Comp. Algebraic Geometry and Commutative Algebra, D. Eisenbud and L. Robbiano (eds.). Symposia Matematica, vol. XXXIV, Istituto Nazionale di Alta Matematica, Cambridge University Press (1993) 216–256.

    Google Scholar 

  37. M. Giusti, J. Heintz and J. Sabia. “On the efficiency of effective Nullstellensätze”. Computational Complexity 3 (1993) 56–95.

    Google Scholar 

  38. M. Giusti, J. Heintz, J.E. Morais, L.M. Pardo. “When polynomial equation systems can be solved’ fast’ ?”. In this volume (1995).

    Google Scholar 

  39. L. Gonzalez Vega. “Determinantal formulae for the solution set of zero-dimensional ideals”. J. of Pure and App. Algebra, 76 (1991) 57–80.

    Google Scholar 

  40. D. Yu. Grigor'ev. “Lower Bounds in Algebraic Computational Complexity”. J. of Soviet Mathematics, 4 (1988) 65–108.

    Google Scholar 

  41. P. Gritzmann and B. Sturmfels. “Minkowski Addition of Polytopes: Computational Complexity and Applications to Gröbner Bases.” SIAM J. Disc. Math., Vol. 6, No. 2 (1993) 246–269.

    Google Scholar 

  42. J. Heintz. “Definability and fast quantifier elimination over algebraically closed fields”. Theor. Comp. Science 24, (1983) 239–278.

    Google Scholar 

  43. J. Heintz. “On the computational complexity of polynomials and bilinear mappings”, Proc. AAECC-5, Springer LNCS 356 (1989) 269–300.

    Google Scholar 

  44. J. Heintz, T. Recio and M.F. Roy. “Algorithms in Real Algebraic Geometry and Applications to Computational Geometry”. DIMACS series in Discrete Math. and Theor. Comp. Sci. 6 (1991) 137–162.

    Google Scholar 

  45. J. Heintz, M.-F. Roy and P. Solerno. “Sur la Complexité du Principe de Tarski-Seidenberg”. Bull. Soc. Math. de France 118, (1990) 101–126.

    Google Scholar 

  46. J. Heintz and J. Morgenstern. “On the intrinsic complexity of elimination theory”. J. of Complexity 9 (1993) 471–498.

    Google Scholar 

  47. J. Heintz and C.P. Schnorr. “Testing Polynomials wich are easy to compute”. In Logic and Algorithmic (an International Symposium in honour of Ernst Specker), Monographie n. 30 de l'Enseignement Mathématique (1982) 237–254. A preliminary version appeared in Proc. 12th Ann. ACM Symp. on Computing (1980) 262–268.

    Google Scholar 

  48. J. Heintz and M. Sieveking. “Lower Bounds for polynomials with algebraic coefficients”, Theoret. Comp. Sci. 11 (1980) 321–330.

    Google Scholar 

  49. G. Hermann. “Die Frage der endlich vielen Schritte in der Theorie der Polynomideale”. Math. Ann. 95 (1926) 736–788.

    Google Scholar 

  50. F. Hurtado and M. Noy. “Ears of triangulations and Catalan numbers”. To appear in Discrete Mathematics.

    Google Scholar 

  51. O.H. Ibarra, S. Moran. “Equivalence of Straight-Line Programs”. J. of the ACM 30 (1983) 217–228.

    Google Scholar 

  52. O.H. Ibarra, S. Moran, E. Rosier. “Probabilistic algorithms and straight-line programs for some rank decision problems”. Inf. Proc. Letters, Vol. 12, No. 5 (1981) 227–232.

    Google Scholar 

  53. E. Kaltofen. “Greatest common divisors of polynomials given by straight-line programs”. J. of the ACM, 35 No. 1 (1988) 234–264.

    Google Scholar 

  54. D.E. Knuth. “The Art of Programming (vol. 2): Semi-numerical Algorithms” Addison-Wesley (1981).

    Google Scholar 

  55. E. Kaltofen. “Factorization of Polynomials given by straight-line programs”. In Randomness in Computation, Advances in Computing Research 5 (1989) 375–412

    Google Scholar 

  56. H. Kobayashi, T. Fujise and A. Furukawa. “Solving systems of algebraic equations equations by general elimination method”. J. Symb. Comp. 5 (1988) 303–320.

    Google Scholar 

  57. J. Kollár. “Sharp Effective Nullstllensatz”, J. of AMS. 1 (1988) 963–975.

    Google Scholar 

  58. T. Krick, Luis M. Pardo. “Une approche informatique pour l'approximation diophantienne”. Comptes Rendues de l'Acad. des Sci. Paris, t. 318, Série I, no. 5, (1994) 407–412.

    Google Scholar 

  59. T. Krick, L.M. Pardo. “A Computational Method for Diophantine Approximation”. To appear in Proc. MEGA'94, Birkhäusser Verlag, (1995).

    Google Scholar 

  60. H.T. Kung. “New algorithms and lower bounds for the parallel evaluation of certain rational expressions and recurrences”. J. of the ACM, 23, (1976) 534–543.

    Google Scholar 

  61. D. Lazard. “Résolution des systèmes d'équations algébriques”, Theor. Comp. Sci. 15 (1981) 77–110.

    Google Scholar 

  62. D. Lazard. “Systems of Algebraic Equations (Algorithms and Complexity)”. In Computational Algebraic Geometry and Commutative Algebra”, Proc. of the Cortona Conf. on Comp. Algebraic Geometry and Commutative Algebra, D. Eisenbud and L. Robbiano (eds.). Symposia Matematica, vol. XXXIV, Istituto Nazionale di Alta Matematica, Cambridge University Press (1993) 216–256.

    Google Scholar 

  63. A.K. Lenstra, H.W. Lenstra. “Algorithms in Number Theory”. In Handbook of Theoretical Computer Science, Elsevier, ch. 12 (1990) 673–715.

    Google Scholar 

  64. M. Li and P.M.B. Vitányi. “Kolmogorov Complexity and its Applications”. In Handbook of Theoretical Computer Science, Elsevier, ch. 4 (1990) 187–254.

    Google Scholar 

  65. D. W. Masser and G. Wüstholz. “Fields of large transcendence degree generated by values of elliptic functions”, Invent. Math. 72 (1971) 407–463.

    Google Scholar 

  66. G. Matera. “Integration of multivariate rationals functions given by straight-line programs”. In this volume (1995).

    Google Scholar 

  67. G. Matera, J.M. Turull. “The complexity of elimination: upper bounds”. Work in preparation (1995).

    Google Scholar 

  68. E. Mayr. “Membership in Polynomial Ideals over 67-02 Is Exponential Space Complete”.

    Google Scholar 

  69. E. Mayr and A. Meyer. “The complexity of the word problem for commutative semigroups“, Advances in Math. 46 (1982) 305–329.

    Google Scholar 

  70. M. Mignotte. “Mathématiques pour le Cacul Formel”, Presses Univ. de France (1989).

    Google Scholar 

  71. J.L. Montaña and L.M. Pardo.“Lower bounds for Arithmetic Networks”. AAECC 4 (1993) 1–24.

    Google Scholar 

  72. J.L. Montaña, J.E. Morais, L.M. Pardo. “Lower Bounds for Arithmetic Networks II:Sum of Betti Numbers”.To appear in AAECC 7 (1995).

    Google Scholar 

  73. T. Mora, L. Robbiano. “Points in Affine and Projective Sapces”. In Computational Algebraic Geometry and Commutative Algebra”, Proc. of the Cortona Conf. on Comp. Algebraic Geometry and Commutative Algebra, D. Eisenbud and L. Robbiano (eds.). Symposia Matematica, vol. XXXIV, Istituto Nazionale di Alta Matematica, Cambridge University Press (1993) 216–256.

    Google Scholar 

  74. K. Mulmuley. “A fast parallel algorithmn to compute the rank of a matrix over an arbitrary field”. COMBINATORICA, 7(1) (1987) 101–104.

    Google Scholar 

  75. Newton,I. “De Analysi per Aequationes Infinitas”. (1667).

    Google Scholar 

  76. A.M. Ostrowski.“On two problems in abstract algebra connected with Horner's rule”. In Studies in Math. and Mech. presented to Richard von Mises Academicv Press (1954) 40–48.

    Google Scholar 

  77. P. Philippon.“Sur des hauteurs alternatives, I”. Math. Ann. 289 (1991) 255–283.

    Google Scholar 

  78. P. Philippon. “Sur des hauteurs alternatives, II”. Ann. Inst. Fourier, Grenoble 44, 2 (1994) 1043–1065.

    Google Scholar 

  79. P. Philippon.“Sur des hauteurs alternatives, III”. To appear in J. Math. Pures Appl.

    Google Scholar 

  80. D.A. Plaisted.“Sparse Complex Polynomials and Polynomial Reducibility”. J. of Comput. Syst. Sci. 14 (1977) 210–221.

    Google Scholar 

  81. R. Pollack, M.F. Roy. “On the number of cells defined by a set of polynomials”. C.R. Acad. Sci. Paris 316 (1991) 573–577.

    Google Scholar 

  82. J. Renegar. “On the Worst-Case Arithmetic Complexity of Approximating Zeros of Polynomials”. J. of Complexity 3 (1987) 90–113

    Google Scholar 

  83. H. Riesel. “Prime Numbers and Computer Methods for Factorization”. Birkhäuser Progress in Math. 57 (1985).

    Google Scholar 

  84. J. Sabia and P. Solernó. “Bounds for traces in complete intersections and degrees in the Nullstellensatz”. To appear in AAECC Journal (1993).

    Google Scholar 

  85. F. Santos. “Geometriá Combinatoria de Curvas Algebraicas y Diagramas de Delaunay en el piano”. Tesis, U. de Cantabria (1995).

    Google Scholar 

  86. J.T. Schwartz. “Fast Probabilistic Algorithms for Verification of Polynomial Identities”. J. of the ACM 27, (1980), 701–717.

    Google Scholar 

  87. T. Schneider. “Einführung in die transzendenten Zahlen”. Springer-Verlag, Berlin (1957).

    Google Scholar 

  88. A. Schonhage, A. F.W. Grotefeld, E. Vetter. “Fast Algorithms (A Multitape Turing machine Implementation)”. B. I. Wissenschaftverlag (1994).

    Google Scholar 

  89. M. Shub, S. Smale. “Complexity of Bézout's theorem I: Geometric aspects”. J. Am. Math. Soc., Vol. 6, No. 2 (1993) 459–501

    Google Scholar 

  90. M. Shub, S. Smale. “Complexity of Bézout's theorem V: Polynomial time”. Theor. Comp. Sci., 133 (1994) 141–164

    Google Scholar 

  91. M. Shub, S.Smale. “On the Intractability of Hilbert's Nullstellensatz and an algebraic version of “NP=P ?”. Preprint of IBM Research Division RC 19624 6/23/94 (1994).

    Google Scholar 

  92. S. Smale. “On the Topology of Algorithms, I”. J. of Complexity, 3 (1987) 81–89.

    Google Scholar 

  93. S. Smale.“On the Efficiency of Algorithms of Analysis”. Bull of the AMS 13, N. 2 (1985) 87–121.

    Google Scholar 

  94. V. Strassen. “Berechnung und Programm I”. Acta Inform., 1 (1972) 320–334.

    Google Scholar 

  95. V. Strassen. “Polynomials with rational coefficients which are hard to compute”, SIAM J. Comput. 3 (1974) 128–149.

    Google Scholar 

  96. V. Strassen. “Vermeidung von Divisionen”, Crelle J. Reine Angew. Math. 264 (1973) 184–202.

    Google Scholar 

  97. V. Strassen. “Algebraic Complexity Theory”. In Handbook of Theoretical Computer Science, Elsevier, ch. 11 (1990) 634–671.

    Google Scholar 

  98. B. Sturmfels. “Some Applications of Affine Gale Diagrams to Polytopes with Few Vertices.” SIAM J. Disc. Math., Vol. 1, No. 1 (1988) 121–133

    Google Scholar 

  99. B. Sturmfels. “Sparse Elimination Theory”. In Computational Algebraic Geometry and Commutative Algebra”, Proc. of the Cortona Conf. on Comp. Algebraic Geometry and Commutative Algebra, D. Eisenbud and L. Robbiano (eds.). Symposia Matematica, vol. XXXIV, Istituto Nazionale di Alta Matematica, Cambridge University Press (1993) 264–298.

    Google Scholar 

  100. B. Sturmfels. “Gröbner Bases of Toric Varieties.” Tohoku Math. J., 43 (1991) 249–261.

    Google Scholar 

  101. A. C.C. Yao. “On Paralell Computation for the Knapsack problem”. J. of the ACM, vol. 29, No. 3 (1982) 898–903.

    Google Scholar 

  102. R. Zippel. “Interpolating Polynomials from their Values”. J. Symbol. Comput., 9 (1990) 375–403

    Google Scholar 

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  1. Depto. de Matemáticas, Estadística y Computación F. de Ciencias, Univ. de Cantabria, E-39071, Santander, Spain

    Luis M. Pardo

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Gérard Cohen Marc Giusti Teo Mora

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Pardo, L.M. (1995). How lower and upper complexity bounds meet in elimination theory. In: Cohen, G., Giusti, M., Mora, T. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1995. Lecture Notes in Computer Science, vol 948. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60114-7_4

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