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DEFORMATION TECHNIQUES 107 [Buc85] B. Buchberger, Gro bner bases: An algorithmic method in polynomial ideal theory, in ``Multidimensional System Theory'' (N. K. Bose et al., Eds.), pp. 374383, Reidel, Dordrecht, 1985. [Can90] J. Canny, Generalised characteristic polynomials, J. Symbolic Comput. 9 (1990), 241250. [Cas97] D. Castro, ``Sobre la complejidad de la aproximacio n diofa ntica y los fundamentos del ana lisis nume rico,'' Master's thesis, Universidad de Cantabria, Santander, Spain, 1997. [CG83] A. L. Chistov and D. Y. Grigoriev, Subexponential time solving systems of algebraic equations, LOMI preprint E-9-83, E-10-83, Steklov Institute, Leningrad, 1983. [Col97] A. Colin, ``The orie des invariants effective,'' Ph.D. thesis, Ecole Polytechnique, Palaiseau-Paris, France, 1997. [Ful84] W. Fulton, `Ìntersection Theory,'' 2nd ed., Ergebnisse der Mathematik, Springer-Verlag, New YorkBerlin, 1984. [GH93] M. Giusti and J. Heintz, La de termination des points isole s et de la dimension d'une varie te alge brique peut se faire en temps polynomial, in ``Computational Algebraic Geometry and Commutative Algebra, Symposia Mathematica'' (D. Eisenbud and L. Robbiano, Eds.), Vol. 34, pp. 216256, Cambridge Univ. Press, Cambridge, UK, 1993. [GHH + 97] M. Giusti, K. Ha gele, J. Heintz, J. E. Morais, J. L. Montan~ a, and L. M. Pardo, Lower bounds for diophantine approximation, J. Pure Appl. Algebra 117118 (1997), 277317. [GHL + 97] M. Giusti, K. Ha gele, G. Lecerf, J. Marchand, and B. Salvy, ``Computing the Dimension of a Projective Variety: The Projective Noether Maple Package,'' Research Report 3224, INRIA, France, July 1997. [GHM + 98] M. Giusti, J. Heintz, J. E. Morais, J. Morgenstern, and L. M. Pardo, Straight-line programs in geometric elimination theory, J. Pure Appl. Algebra 124 (1998), 101146. [GHMP97] M. Giusti, J. Heintz, J. E. Morais, and L. M. Pardo, Le ro^ le des structures de donne es dans les proble mes d'e limination, C. R. Acad. Sci. Paris 325 (1997), 12231228. [GM89] P. Gianni and T. Mora, Algebraic solution of systems of polynomial equations using Gro bner bases, in ``Proceedings AAECC-5,'' Lecture Notes in Comput. Sci., Vol. 356, pp. 247257, Springer-Verlag, New YorkBerlin, 1989. [GV88] D. Yu. Grigoriev and N. N. Vorobjov, Jr., Solving systems of polynomial inequalities in sub-exponential time, J. Symbolic Comput. 5 (1988), 3764. [Hei83] J. Heintz, Definability and fast quantifier elimination in algebraically closed fields, Theoret. Comput. Sci. 24 (1983), 239277. [HMPS98] K. Ha gele, J. E. Morais, L. M. Pardo, and M. Sombra, On the intrinsic complexity of the arithmetic Nullstellensatz, J. Pure Appl. Algebra, in press. [HRS90] J. Heintz, M.-F. Roy, and P. Solerno , Sur la complexite du principe de TarskiSeidenberg, Bull. Soc. Math. France 118 (1990), 101126. [Ive73] B. Iversen, Generic local structure of the morphisms in commutative algebra, in ``Lecture Notes in Math.,'' Vol. 310, Springer-Verlag, New YorkBerlin, 1973. [Kal85] E. Kaltofen, Computing with polynomials given by straight-line programs. I. Greatest common divisors, in ``Proceedings of the 17th Ann. ACM Symposium
108 HEINTZ ET AL. on Theory of Computing, Providence, RI,'' pp. 131142, ACM Press, New York, 1985. [Koe03] J. Koenig, `Èinleitung in die allgemeine Theorie der algebraischen Gro ssen,'' Teubner, Leipzig, 1903. [Kov90] P. Kovacs, Minimum degree solutions for the inverse kinematics problem by application of the Buchberger algorithm, in ``Proceedings of the 2nd Int. Workshop on Advances in Robot Kinematics, University of Linz, Austria, September 1012, 1990,'' Springer-Verlag, New YorkBerlin, 1990. [KP96] T. Krick and L. M. Pardo, A computational method for diophantine approximation, in `Àlgorithms in Algebraic Geometry and Applications, Proceedings of MEGA'94'' (L. Gonza lez-Vega and T. Recio, Eds.), Progress in Mathematics, Vol. 143, pp. 193254, Birkha user Verlag, Basel, 1996. [Kro82] L. Kronecker, Grundzu ge einer arithmetischen Theorie de algebraischen Gro ssen, J. Reine Angew. Math. 92 (1882), 122. [LV98] M. J. Gonzalez Lopez and L. Gonzalez Vega, Newton identities in the multivariate case: Pham systems, in ``London Mathematical Society Lecture Notes Series,'' Vol. 251, pp. 351366, Cambridge Univ. Press, Cambridge, UK, 1998. [Mac16] F. S. Macaulay, `Àlgebraic Theory of Module Systems,'' Cambridge Univ. Press, Cambridge, UK, 1916. [Mat97] G. Matera, ``Sobre la complejidad en espacio y tiempo de la eliminacio n geome trica,'' Ph.D. thesis, Universidad de Buenos Aires, Argentina, 1997. [Mis93] B. Mishra, `Àlgorithmic Algebra,'' Springer-Verlag, New York, 1993. [Mor97] J. E. Morais, ``Resolucio n eficaz de sistemas de ecuaciones polinomiales,'' Ph.D. thesis, Universidad de Cantabria, Santander, Spain, 1997. [MP97] B. Mourrain and V. Pan, Solving special polynomial systems by using structural matrices and algebraic residues, in ``Proceedings Foundations of Computational Mathematics, FOCM'97'' (F. Cucker and M. Shub, Eds.), pp. 287304, Springer- Verlag, New YorkBerlin, 1997. [Nar66] R. Narasimhan, Introduction to the theory of analytic spaces, in ``Lecture Notes in Math.,'' Vol. 25, Springer-Verlag, New YorkBerlin, 1966. [Ren92] J. Renegar, On the computational complexity and geometry of the first-order theory of the reals. Part I. Introduction, preliminaries, the geometry of semialgebraic sets, the decision problem for the existential theory of the reals, J. Symbolic Comput. 13 (1992), 255299. [Sch95] J. Schmid, On the affine Be zout inequality, Manuscripta Math. 88 (1995), 225 232. [SGV94] A. Scho nhage, F. W. Grotefeld, and E. Vetter, ``Fast Algorithms: A Multitape Turing Machine Implementation,'' Wissenschaftsverlag, Mannheim, 1994. [Som97] M. Sombra, Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz, J. Pure Appl. Algebra 117118 (1997), 565599. [SS93a] M. Shub and S. Smale, Complexity of Be zout's theorem. I. Geometric aspects, J. Amer. Math. Soc. 6 (1993), 459501. [SS93b] M. Shub and S. Smale, Complexity of Be zout's theorem. II. Volumes and probabilities, in ``Proceedings of MEGA'92,'' Progress in Mathematics, Vol. 109, pp. 267285, Birkha user Verlag, Basel, 1993. [SS93c] M. Shub and S. Smale, Complexity of Be zout's theorem. III. Condition number and packing, J. Complexity 9 (1993), 414.
Page 1 and 2: journal of complexity 16, 70109 (20
Page 3 and 4: 72 HEINTZ ET AL. In this contributi
Page 5 and 6: 74 HEINTZ ET AL. affine varieties ?
Page 7 and 8: 76 HEINTZ ET AL. holds (here *? &1
Page 9 and 10: 78 HEINTZ ET AL. A labeled node is
Page 11 and 12: 80 HEINTZ ET AL. Given a parameter
Page 13 and 14: 82 HEINTZ ET AL. respect to the bas
Page 15 and 16: 84 HEINTZ ET AL. The zero dimension
Page 17 and 18: 86 HEINTZ ET AL. Let F(X):=(F 1 (T,
Page 19 and 20: 88 HEINTZ ET AL. Finally, plugging
Page 21 and 22: 90 HEINTZ ET AL. Q approximates Q w
Page 23 and 24: 92 HEINTZ ET AL. Now, recall from (
Page 25 and 26: 94 HEINTZ ET AL. Proof. All the pol
Page 27 and 28: 96 HEINTZ ET AL. Taking into accoun
Page 29 and 30: 98 HEINTZ ET AL. By means of the we
Page 31 and 32: 100 HEINTZ ET AL. Applying the algo
Page 33 and 34: 102 HEINTZ ET AL. time economy is o
Page 35 and 36: 104 HEINTZ ET AL. in Q[T, X] and, t
Page 37: 106 HEINTZ ET AL. P 1 , ..., P n yi

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