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Volumn 17, Issue 1, 2001, Pages 212-303

Kronecker's and Newton's approaches to solving: A first comparison

Author keywords

Approximate zero; Degree of algebraic varieties; Height of diophantine varieties; Kronecker's solution; Newton operator; Straight line programs; Turing machine complexity

Indexed keywords


EID: 0035294748     PISSN: 0885064X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcom.2000.0572     Document Type: Article
Times cited : (22)

References (108)
  • 2
  • 3
    • 84966219173 scopus 로고
    • Elliptic curves and primality proving
    • Atkin A. O. L., Morain F. Elliptic curves and primality proving. Math. Comp. 61:1993;29-68.
    • (1993) Math. Comp. , vol.61 , pp. 29-68
    • Atkin, A.O.L.1    Morain, F.2
  • 4
    • 0003335813 scopus 로고
    • Structural Complexity, I
    • New York/Berlin: Springer-Verlag
    • Balcázar J. L., Díaz J., Gabarró J. Structural Complexity, I. EATCS. 11:1988;Springer-Verlag, New York/Berlin.
    • (1988) EATCS , vol.11
    • Balcázar, J.L.1    Díaz, J.2    Gabarró, J.3
  • 5
    • 34250406508 scopus 로고
    • The number of roots of a system of equations
    • Bernstein D. N. The number of roots of a system of equations. Funktsional Anal. i Prilozhen. 9:1975;1-4.
    • (1975) Funktsional Anal. I Prilozhen. , vol.9 , pp. 1-4
    • Bernstein, D.N.1
  • 11
    • 0020749812 scopus 로고
    • Improving an algorithm for factoring polynomials over a finite field and constructing large irreducible polynomials
    • Camion P. F. Improving an algorithm for factoring polynomials over a finite field and constructing large irreducible polynomials. IEEE Trans. Inform. Theory. 29:1983;378-385.
    • (1983) IEEE Trans. Inform. Theory , vol.29 , pp. 378-385
    • Camion, P.F.1
  • 12
    • 84966220369 scopus 로고
    • A new algorithm for factoring polynomials over finite fields
    • Cantor D. G., Zassenhaus H. A new algorithm for factoring polynomials over finite fields. Math. Comp. 36:1981;587-592.
    • (1981) Math. Comp. , vol.36 , pp. 587-592
    • Cantor, D.G.1    Zassenhaus, H.2
  • 16
    • 0007468723 scopus 로고    scopus 로고
    • Newton's method for overdetermined systems of equations
    • in press
    • J. P. Dedieu, and, M. Shub, Newton's method for overdetermined systems of equations, Math. Comp, in press.
    • Math. Comp
    • Dedieu, J.P.1    Shub, M.2
  • 17
    • 0001049080 scopus 로고    scopus 로고
    • Approximate solutions of numerical problems, condition number analysis and condition number theorem
    • Providence: Amer. Math. Soc. p. 263-283
    • Dedieu J. P. Approximate solutions of numerical problems, condition number analysis and condition number theorem. The Mathematics of Numerical Analysis, Park City, UT, 1995. 1996;Amer. Math. Soc. Providence. p. 263-283.
    • (1996) The Mathematics of Numerical Analysis, Park City, UT, 1995
    • Dedieu, J.P.1
  • 19
    • 0040250392 scopus 로고    scopus 로고
    • Condition operators, condition numbers, and condition number theory for the generalized eigenvalue problem
    • Dedieu J. P. Condition operators, condition numbers, and condition number theory for the generalized eigenvalue problem. Linear Algebra Appl. 263:1997;1-24.
    • (1997) Linear Algebra Appl. , vol.263 , pp. 1-24
    • Dedieu, J.P.1
  • 20
    • 0031539451 scopus 로고    scopus 로고
    • Estimations for the separation number of a polynomial system
    • Dedieu J. P. Estimations for the separation number of a polynomial system. J. Symbolic Comput. 24:1997;683-693.
    • (1997) J. Symbolic Comput. , vol.24 , pp. 683-693
    • Dedieu, J.P.1
  • 21
    • 0007547104 scopus 로고    scopus 로고
    • On simple double zeros and badly conditioned zeros of analytic functions of n variables
    • J. P. Dedieu, and, M. Shub, On simple double zeros and badly conditioned zeros of analytic functions of n variables, Math. Comp.
    • Math. Comp.
    • Dedieu, J.P.1    Shub, M.2
  • 22
    • 0001577527 scopus 로고    scopus 로고
    • Some lower bounds for the complexity of continuation methods
    • Dedieu J. P., Smale S. Some lower bounds for the complexity of continuation methods. J. Complexity. 14:1998;454-465.
    • (1998) J. Complexity , vol.14 , pp. 454-465
    • Dedieu, J.P.1    Smale, S.2
  • 24
    • 0001380055 scopus 로고
    • Exact solution of linear equations using p-adic expansions
    • Dixon J. Exact solution of linear equations using p-adic expansions. Numer. Math. 40:1982;137-141.
    • (1982) Numer. Math. , vol.40 , pp. 137-141
    • Dixon, J.1
  • 26
    • 0000802374 scopus 로고
    • The approximation of one matrix by another of lower rank
    • Eckardt C., Young G. The approximation of one matrix by another of lower rank. Psychometrika. 1:1936;211-218.
    • (1936) Psychometrika , vol.1 , pp. 211-218
    • Eckardt, C.1    Young, G.2
  • 28
    • 0003314349 scopus 로고
    • Intersection Theory
    • New York/Berlin: Springer-Verlag
    • Fulton W. Intersection Theory. Ergebnisse der Mathematik. 3:1984;Springer-Verlag, New York/Berlin.
    • (1984) Ergebnisse Der Mathematik , vol.3
    • Fulton, W.1
  • 30
    • 0000750175 scopus 로고
    • Algorithmes - Disons rapides - Pour la décomposition d'une variété algébrique en composantes irréductibles et équidimensionelles
    • T. Mora, & C. Traverso. Basel: Birkhäuser Progress in Mathematics
    • Giusti M., Heintz J. Algorithmes - disons rapides - pour la décomposition d'une variété algébrique en composantes irréductibles et équidimensionelles. Mora T., Traverso C. Proceedings of MEGA'90. Progress in Mathematics. 94:1991;169-194 Birkhäuser, Basel.
    • (1991) Proceedings of MEGA'90 , vol.94 , pp. 169-194
    • Giusti, M.1    Heintz, J.2
  • 31
    • 0000936391 scopus 로고
    • La détermination des points isolés et de la dimension d'une variété algébrique peut se faire en temps polynomial
    • D. Eisenbud, & L. Robbiano. Cambridge: Cambridge Univ. Press
    • Giusti M., Heintz J. La détermination des points isolés et de la dimension d'une variété algébrique peut se faire en temps polynomial. Eisenbud D., Robbiano L. Computational Algebraic Geometry and Commutative Algebra. Symposia Matematica. 34:1993;216-256 Cambridge Univ. Press, Cambridge.
    • (1993) Computational Algebraic Geometry and Commutative Algebra. Symposia Matematica , vol.34 , pp. 216-256
    • Giusti, M.1    Heintz, J.2
  • 34
    • 0031539424 scopus 로고    scopus 로고
    • Le rôle des structures de données dans les problèmes d'élimination
    • Giusti M., Heintz J., Morais J. E., Pardo L. M. Le rôle des structures de données dans les problèmes d'élimination. C.R. Acad. Sci. Paris. 325:1997;1223-1228.
    • (1997) C.R. Acad. Sci. Paris , vol.325 , pp. 1223-1228
    • Giusti, M.1    Heintz, J.2    Morais, J.E.3    Pardo, L.M.4
  • 35
    • 2042528768 scopus 로고    scopus 로고
    • Solving some over-determined systems
    • in press
    • M. Giusti, and, E. Schost, Solving some over-determined systems, in, Proc. ISSAC, 1999, in press.
    • (1999) In Proc. ISSAC
    • Giusti, M.1    Schost, E.2
  • 39
    • 0000627865 scopus 로고
    • Definability and fast quantifier elimination in algebraically closed fields
    • Heintz J. Definability and fast quantifier elimination in algebraically closed fields. Theoret. Comput. Sci. 24:1983;239-277.
    • (1983) Theoret. Comput. Sci. , vol.24 , pp. 239-277
    • Heintz, J.1
  • 42
    • 84976819141 scopus 로고
    • Equivalence of straight-line programs
    • Ibarra O. H., Moran S. Equivalence of straight-line programs. J. Assoc. Comput. Mach. 30:1983;217-228.
    • (1983) J. Assoc. Comput. Mach. , vol.30 , pp. 217-228
    • Ibarra, O.H.1    Moran, S.2
  • 46
    • 0007513183 scopus 로고
    • Polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers
    • Kannan R., Lenstra A. K., Lovász L. Polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers. Math. Comp. 50:1988;235-250.
    • (1988) Math. Comp. , vol.50 , pp. 235-250
    • Kannan, R.1    Lenstra, A.K.2    Lovász, L.3
  • 48
    • 84968511007 scopus 로고
    • On approximate zeros and rootfinding algorithms for a complex polynomial
    • Kim M.-H. On approximate zeros and rootfinding algorithms for a complex polynomial. Math. Comp. 51:1988;707-719.
    • (1988) Math. Comp. , vol.51 , pp. 707-719
    • Kim, M.-H.1
  • 49
    • 38249005379 scopus 로고
    • Topological complexity of a root finding algorithm
    • Kim M.-H. Topological complexity of a root finding algorithm. J. Complexity. 5:1989;331-344.
    • (1989) J. Complexity , vol.5 , pp. 331-344
    • Kim, M.-H.1
  • 51
    • 0001379976 scopus 로고
    • Une approche informatique pour l'approximation diophantienne
    • Krick T., Pardo L. M. Une approche informatique pour l'approximation diophantienne. C.R. Acad. Sci. Paris. 318:1994;407-412.
    • (1994) C.R. Acad. Sci. Paris , vol.318 , pp. 407-412
    • Krick, T.1    Pardo, L.M.2
  • 52
    • 0000697864 scopus 로고    scopus 로고
    • A computational method for diophantine approximation
    • L. González-Vega, & T. Recio. Progress in Mathematics Basel: Birkhäuser
    • Krick T., Pardo L. M. A computational method for diophantine approximation. González-Vega L., Recio T. Algorithms in Algebraic Geometry and Applications, Proceedings of MEGA'94. Progress in Mathematics. 143:1996;193-254 Birkhäuser, Basel.
    • (1996) Algorithms in Algebraic Geometry and Applications, Proceedings of MEGA'94 , vol.143 , pp. 193-254
    • Krick, T.1    Pardo, L.M.2
  • 53
    • 84941839155 scopus 로고
    • Grundzüge einer arithmetischen theorie de algebraischen grössen
    • Kronecker L. Grundzüge einer arithmetischen theorie de algebraischen grössen. J. Reine Angew. Math. 92:1882;1-122.
    • (1882) J. Reine Angew. Math. , vol.92 , pp. 1-122
    • Kronecker, L.1
  • 55
    • 0022014061 scopus 로고
    • Factoring polynomials over algebraic number fields
    • Landau S. Factoring polynomials over algebraic number fields. SIAM J. Comput. 14:1985;184-195.
    • (1985) SIAM J. Comput. , vol.14 , pp. 184-195
    • Landau, S.1
  • 56
    • 0020733275 scopus 로고
    • Solvability by radicals is in polynomial time
    • Landau S., Miller G. L. Solvability by radicals is in polynomial time. J. Comput. System Sci. 30:1985;179-208.
    • (1985) J. Comput. System Sci. , vol.30 , pp. 179-208
    • Landau, S.1    Miller, G.L.2
  • 59
    • 85031899865 scopus 로고
    • Factoring polynomials over algebraic number fields
    • Berlin: Springer-Verlag. p. 245-254
    • Lenstra A. K. Factoring polynomials over algebraic number fields. Computer Algebra, London, 1983. 1983;Springer-Verlag, Berlin. p. 245-254.
    • (1983) Computer Algebra, London, 1983
    • Lenstra, A.K.1
  • 62
    • 34250244723 scopus 로고
    • Factoring polynomials with rational coefficients
    • Lenstra A. K., Lenstra H. W. Jr., Lovász L. Factoring polynomials with rational coefficients. Math. Ann. 261:1982;513-534.
    • (1982) Math. Ann. , vol.261 , pp. 513-534
    • Lenstra, A.K.1    Lenstra H.W., Jr.2    Lovász, L.3
  • 65
    • 0028516907 scopus 로고
    • On generalized newton algorithms: Quadratic convergence, path-following and error analysis
    • Malajovich G. On generalized newton algorithms: quadratic convergence, path-following and error analysis. Theoret. Comput. Sci. 133:1994;65-84.
    • (1994) Theoret. Comput. Sci. , vol.133 , pp. 65-84
    • Malajovich, G.1
  • 71
    • 0002525647 scopus 로고
    • The non-scalar model of complexity in computational geometry
    • C. Traverso, & T. Mora. Progress in Mathematics Basel: Birkhäuser
    • Montaña J. L., Pardo L. M., Recio T. The non-scalar model of complexity in computational geometry. Traverso C., Mora T. Effective Methods in Algebraic Geometry, Proceedings of MEGA'90. Progress in Mathematics. 94:1991;347-361 Birkhäuser, Basel.
    • (1991) Effective Methods in Algebraic Geometry, Proceedings of MEGA'90 , vol.94 , pp. 347-361
    • Montaña, J.L.1    Pardo, L.M.2    Recio, T.3
  • 73
    • 25044436578 scopus 로고
    • Elliptic curves, primality proving and some titanic primes
    • Morain F. Elliptic curves, primality proving and some titanic primes. Astérisque. 198-200:1992;245-251.
    • (1992) Astérisque , vol.198-200 , pp. 245-251
    • Morain, F.1
  • 75
    • 84955604612 scopus 로고
    • How lower and upper complexity bounds meet in elimination theory
    • G. Cohen, H. Giusti, & T. Mora. Lecture Notes in Computer Science Berlin: Springer-Verlag
    • Pardo L. M. How lower and upper complexity bounds meet in elimination theory. Cohen G., Giusti H., Mora T. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Paris, 1995. Lecture Notes in Computer Science. 948:1995;33-69 Springer-Verlag, Berlin.
    • (1995) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Paris, 1995 , vol.948 , pp. 33-69
    • Pardo, L.M.1
  • 76
    • 0000572440 scopus 로고
    • Sur des hauteurs alternatives, I
    • Philippon P. Sur des hauteurs alternatives, I. Math. Ann. 289:1991;255-283.
    • (1991) Math. Ann. , vol.289 , pp. 255-283
    • Philippon, P.1
  • 77
    • 0000924409 scopus 로고
    • Sur des hauteurs alternatives, II
    • Philippon P. Sur des hauteurs alternatives, II. Ann. Inst. Fourier (Grenoble). 44:1994;1043-1065.
    • (1994) Ann. Inst. Fourier (Grenoble) , vol.44 , pp. 1043-1065
    • Philippon, P.1
  • 78
    • 0000801815 scopus 로고
    • Sur des hauteurs alternatives, III
    • Philippon P. Sur des hauteurs alternatives, III. J. Math. Pures Appl. 74:1995;345-365.
    • (1995) J. Math. Pures Appl. , vol.74 , pp. 345-365
    • Philippon, P.1
  • 80
    • 38249035936 scopus 로고
    • On the worst-case arithmetic complexity of approximating zeros of polynomials
    • Renegar J. On the worst-case arithmetic complexity of approximating zeros of polynomials. J. Complexity. 3:1987;90-113.
    • (1987) J. Complexity , vol.3 , pp. 90-113
    • Renegar, J.1
  • 87
    • 0000960921 scopus 로고
    • Computational complexity: On the geometry of polynomials and a theory of cost, I
    • Shub M., Smale S. Computational complexity: On the geometry of polynomials and a theory of cost, I. Ann. Sci. École Norm. Sup. 18:1985;107-142.
    • (1985) Ann. Sci. École Norm. Sup. , vol.18 , pp. 107-142
    • Shub, M.1    Smale, S.2
  • 88
    • 0022659074 scopus 로고
    • Computational complexity: On the geometry of polynomials and a theory of cost, II
    • Shub M., Smale S. Computational complexity: On the geometry of polynomials and a theory of cost, II. SIAM J. Comput. 15:1986;145-161.
    • (1986) SIAM J. Comput. , vol.15 , pp. 145-161
    • Shub, M.1    Smale, S.2
  • 89
    • 84968505377 scopus 로고
    • Complexity of Bézout's theorem. I. Geometric aspects
    • Shub M., Smale S. Complexity of Bézout's theorem. I. Geometric aspects. J. Amer. Math. Soc. 6:1993;459-501.
    • (1993) J. Amer. Math. Soc. , vol.6 , pp. 459-501
    • Shub, M.1    Smale, S.2
  • 91
    • 0002102930 scopus 로고
    • Complexity of Bézout's theorem. III. Condition number and packing
    • Shub M., Smale S. Complexity of Bézout's theorem. III. Condition number and packing. J. Complexity. 9:1993;4-14.
    • (1993) J. Complexity , vol.9 , pp. 4-14
    • Shub, M.1    Smale, S.2
  • 92
    • 0000675799 scopus 로고    scopus 로고
    • Complexity of Bézout's theorem. IV. Probability of success, extensions
    • in press
    • M. Shub, and, S. Smale, Complexity of Bézout's theorem. IV. Probability of success, extensions, SIAM J. Numer. Anal, in press.
    • SIAM J. Numer. Anal
    • Shub, M.1    Smale, S.2
  • 93
    • 0028516294 scopus 로고
    • Complexity of Bézout's theorem. V. Polynomial time
    • Shub M., Smale S. Complexity of Bézout's theorem. V. Polynomial time. Theoret. Comput. Sci. 133:1994;141-164.
    • (1994) Theoret. Comput. Sci. , vol.133 , pp. 141-164
    • Shub, M.1    Smale, S.2
  • 94
    • 0000675799 scopus 로고    scopus 로고
    • Complexity of Bezout's theorem. IV. Probability of success and extensions
    • Shub M., Smale S. Complexity of Bezout's theorem. IV. Probability of success and extensions. SIAM J. Numer. Anal. 33:1996;128-148.
    • (1996) SIAM J. Numer. Anal. , vol.33 , pp. 128-148
    • Shub, M.1    Smale, S.2
  • 95
    • 84966209344 scopus 로고
    • The fundamental theorem of algebra and complexity theory
    • Smale S. The fundamental theorem of algebra and complexity theory. Bull. Amer. Math. Soc. (N.S.). 4:1981;1-36.
    • (1981) Bull. Amer. Math. Soc. (N.S.) , vol.4 , pp. 1-36
    • Smale, S.1
  • 96
    • 84967770163 scopus 로고
    • On the efficiency of algorithms of analysis
    • Smale S. On the efficiency of algorithms of analysis. Bull. Amer. Math. Soc. 13:1985;87-121.
    • (1985) Bull. Amer. Math. Soc. , vol.13 , pp. 87-121
    • Smale, S.1
  • 101
    • 0003306410 scopus 로고    scopus 로고
    • Gröbner Bases and Convex Polytopes
    • Providence: Amer. Math. Soc.
    • Sturmfels B. Gröbner Bases and Convex Polytopes. University Lecture Series. 8:1996;Amer. Math. Soc. Providence.
    • (1996) University Lecture Series , vol.8
    • Sturmfels, B.1
  • 103
    • 0003539886 scopus 로고
    • New York/Berlin: Springer-VerlagTata Institute of Fundamental Research
    • Vogel W. Results on Bézout's Theorem. 1984;Springer-VerlagTata Institute of Fundamental Research, New York/Berlin.
    • (1984) Results on Bézout's Theorem
    • Vogel, W.1
  • 104
    • 0007514270 scopus 로고
    • Une constante universelle pour la convergence de la méthode de Newton
    • Yakoubsohn J. C. Une constante universelle pour la convergence de la méthode de Newton. C.R. Acad. Sci. Paris Sér. I Math. 320:1995;385-390.
    • (1995) C.R. Acad. Sci. Paris Sér. I Math. , vol.320 , pp. 385-390
    • Yakoubsohn, J.C.1
  • 105
    • 33747920150 scopus 로고
    • A universal constant for the convergence of Newton's method and an application to the classical homotopy method
    • Yakoubsohn J. C. A universal constant for the convergence of Newton's method and an application to the classical homotopy method. Numer. Algorithms. 9:1995;223-244.
    • (1995) Numer. Algorithms , vol.9 , pp. 223-244
    • Yakoubsohn, J.C.1
  • 106
  • 108
    • 84925463455 scopus 로고
    • Probabilistic algorithms for sparse polynomials
    • New York/Berlin: Springer-Verlag. p. 216-226
    • Zippel R. Probabilistic algorithms for sparse polynomials. Proceedings EUROSAM'79. Lecture Notes in Comput. Sci. 72:1979;Springer-Verlag, New York/Berlin. p. 216-226.
    • (1979) Proceedings EUROSAM'79 Lecture Notes in Comput. Sci. , vol.72
    • Zippel, R.1


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