메뉴 건너뛰기




Volumn 315, Issue 2-3, 2004, Pages 335-369

Polynomial equation solving by lifting procedures for ramified fibers

Author keywords

Efficient polynomial equation solving; Newton Hensel lifting; Puiseux expansions of space curves; Ramified fibers of dominant mappings

Indexed keywords

ALGEBRA; ALGORITHMS; ITERATIVE METHODS; PARAMETER ESTIMATION; PROBLEM SOLVING; THEOREM PROVING; VECTORS;

EID: 2042544102     PISSN: 03043975     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.tcs.2004.01.015     Document Type: Article
Times cited : (12)

References (68)
  • 3
    • 0010994463 scopus 로고
    • Local parametrizations of space curves at singular points
    • I. Herman, C. Pienovi (Eds.), Computer Graphics and Mathematics, Springer, Berlin Heidelberg, New York
    • M. Alonso, G. Niesi, T. Mora, M. Raimondo, Local parametrizations of space curves at singular points, in: I. Herman, C. Pienovi (Eds.), Computer Graphics and Mathematics, Springer Eurographic Seminar Series, Springer, Berlin Heidelberg, New York, 1991, pp. 61-90.
    • (1991) Springer Eurographic Seminar Series , pp. 61-90
    • Alonso, M.1    Niesi, G.2    Mora, T.3    Raimondo, M.4
  • 4
    • 0002259485 scopus 로고
    • An algorithm for computing analytic branches of space curves at singular points
    • W.-T. Wu, M.-D. Cheng (Eds.), Int. Academic Pub.
    • M. Alonso, G. Niesi, T. Mora, M. Raimondo, An algorithm for computing analytic branches of space curves at singular points, in: W.-T. Wu, M.-D. Cheng (Eds.), Proc. 1992 International Workshop on Mathematical Mechanization, Int. Academic Pub., 1992, pp. 135-166.
    • (1992) Proc. 1992 International Workshop on Mathematical Mechanization , pp. 135-166
    • Alonso, M.1    Niesi, G.2    Mora, T.3    Raimondo, M.4
  • 5
    • 0010270907 scopus 로고
    • Lectures on deformation of singularities
    • M. Artin, Lectures on deformation of singularities, Tata Inst. Fund. Res., 1976.
    • (1976) Tata Inst. Fund. Res.
    • Artin, M.1
  • 6
    • 0002128307 scopus 로고
    • What can be computed in algebraic geometry?
    • D. Eisenbud, L. Robbiano (Eds.), Cambridge University Press, Cambridge
    • D. Bayer, D. Mumford, What can be computed in algebraic geometry? in: D. Eisenbud, L. Robbiano (Eds.), Computational Algebraic Geometry and Commutative Algebra, Symp. Mathematics, XXXIV, Cambridge University Press, Cambridge, 1993, pp. 1-49.
    • (1993) Computational Algebraic Geometry and Commutative Algebra, Symp. Mathematics , vol.34 , pp. 1-49
    • Bayer, D.1    Mumford, D.2
  • 7
    • 0003204425 scopus 로고
    • Gröbner bases, A Computational Approach to Commutative Algebra
    • Springer, Berlin, Heidelberg, New York
    • T. Becker, V. Weispfenning, Gröbner bases, A Computational Approach to Commutative Algebra, Springer Graduate Texts in Mathematics, Vol. 141, Springer, Berlin, Heidelberg, New York, 1993.
    • (1993) Springer Graduate Texts in Mathematics , vol.141
    • Becker, T.1    Weispfenning, V.2
  • 12
    • 2042499126 scopus 로고    scopus 로고
    • Lifting procedures for ramified fibers and polynomial equation solving
    • M. Frías, J. Heintz (Eds.), Buenos Aires, September 2001, Anales JAIIO, Buenos Aires
    • A. Bompadre, G. Matera, R. Wachenchauzer, A. Waissbein, Lifting procedures for ramified fibers and polynomial equation solving, in: M. Fri;́as, J. Heintz (Eds.), Proc. Workshop Argentino de Informática Teórica, WAIT'01, Buenos Aires, September 2001, Vol. 30 Anales JAIIO, Buenos Aires, 2001, pp. 13-31.
    • (2001) Proc. Workshop Argentino de Informática Teórica, WAIT'01 , vol.30 , pp. 13-31
    • Bompadre, A.1    Matera, G.2    Wachenchauzer, R.3    Waissbein, A.4
  • 13
    • 23044521433 scopus 로고    scopus 로고
    • Blow-up vs. spurious steady solutions
    • Bonder J.F., Rossi J. Blow-up vs. spurious steady solutions. Proc. Amer. Math. Soc. 129(1):2001;139-144.
    • (2001) Proc. Amer. Math. Soc. , vol.129 , Issue.1 , pp. 139-144
    • Bonder, J.F.1    Rossi, J.2
  • 14
    • 0000797215 scopus 로고
    • Time space tradeoffs (getting closer to the barriers?)
    • 4th Internat. Symp. on Algorithms and Computation, ISAAC '93, Hong Kong, December 1993, Springer, Berlin
    • A. Borodin, Time space tradeoffs (getting closer to the barriers?), in: 4th Internat. Symp. on Algorithms and Computation, ISAAC '93, Hong Kong, December 1993, Lecture Notes in Computer Science, Vol. 762, Springer, Berlin, 1993, pp. 209-220.
    • (1993) Lecture Notes in Computer Science , vol.762 , pp. 209-220
    • Borodin, A.1
  • 17
    • 0002634193 scopus 로고
    • Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions
    • Chipot M., Fila M., Quittner P. Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions. Acta Math. Univ. Comenianae. 60(1):1991;35-103.
    • (1991) Acta Math. Univ. Comenianae , vol.60 , Issue.1 , pp. 35-103
    • Chipot, M.1    Fila, M.2    Quittner, P.3
  • 20
    • 0003352688 scopus 로고    scopus 로고
    • Using Algebraic Geometry
    • Springer, Berlin, Heidelberg, New York
    • D. Cox, J. Little, D. O'Shea, Using Algebraic Geometry, Graduate Texts in Mathematics, Vol. 185, Springer, Berlin, Heidelberg, New York, 1998.
    • (1998) Graduate Texts in Mathematics , vol.185
    • Cox, D.1    Little, J.2    O'Shea, D.3
  • 21
    • 84857846043 scopus 로고    scopus 로고
    • Applications of computational algebraic geometry
    • AMS, Providence, RI
    • D. Cox, B. Sturmfels, Applications of computational algebraic geometry, Vol. 53, Proc. Symp. Applied Mathematics, AMS, Providence, RI, 1998.
    • (1998) Proc. Symp. Applied Mathematics , vol.53
    • Cox, D.1    Sturmfels, B.2
  • 22
    • 0001731749 scopus 로고
    • Algebraic varieties and schemes
    • I. Shafarevich. Algebraic Geometry I. Berlin: Springer
    • Danilov V. Algebraic varieties and schemes. Shafarevich I. Algebraic Geometry I. Encyclopedia Mathematics Sciences. Vol. 23:1994;167-297 Springer, Berlin.
    • (1994) Encyclopedia Mathematics Sciences , vol.23 , pp. 167-297
    • Danilov, V.1
  • 23
    • 0000906145 scopus 로고
    • Rational Puiseux expansions
    • Duval D. Rational Puiseux expansions. Compositio Math. 70:1989;119-154.
    • (1989) Compositio Math. , vol.70 , pp. 119-154
    • Duval, D.1
  • 24
    • 0003243860 scopus 로고
    • Commutative Algebra with a View Toward Algebraic Geometry
    • Springer, Berlin, Heidelberg, New York
    • D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, Vol. 150, Springer, Berlin, Heidelberg, New York, 1995.
    • (1995) Graduate Texts in Mathematics , vol.150
    • Eisenbud, D.1
  • 25
    • 0036000696 scopus 로고    scopus 로고
    • Numerical blow-up for a nonlinear problem with a nonlinear boundary condition
    • Ferreira R., Groisman P., Rossi J. Numerical blow-up for a nonlinear problem with a nonlinear boundary condition. Math. Models Methods Appl. Sci. 12(4):2002;461-484.
    • (2002) Math. Models Methods Appl. Sci. , vol.12 , Issue.4 , pp. 461-484
    • Ferreira, R.1    Groisman, P.2    Rossi, J.3
  • 26
    • 0004233493 scopus 로고
    • Springer, Berlin, Heidelberg, New York
    • W. Fulton, Intersection Theory, Springer, Berlin, Heidelberg, New York, 1984.
    • (1984) Intersection Theory
    • Fulton, W.1
  • 27
    • 85034753633 scopus 로고
    • Algebraic solution of systems of polynomial equations using Gröbner bases
    • L. Huguet, A. Poli (Eds.), Proc. AAECC-5, Menorca, Spain, June, Springer, Berlin
    • P. Gianni, T. Mora, Algebraic solution of systems of polynomial equations using Gröbner bases, in: L. Huguet, A. Poli (Eds.), Proc. AAECC-5, Menorca, Spain, June 1987, Lecture Notes in Computer Science, Vol. 356, Springer, Berlin, 1989, pp. 247-257.
    • (1987) Lecture Notes in Computer Science , vol.356 , pp. 247-257
    • Gianni, P.1    Mora, T.2
  • 29
    • 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 (Eds.), Computational Algebraic Geometry and Commutative Algebra, Cambridge University Press, Cambridge, MA
    • M. Giusti, J. Heintz, La détermination des points isolés et de la dimension d'une variété algébrique peut se faire en temps polynomial, in: D. Eisenbud, L. Robbiano (Eds.), Computational Algebraic Geometry and Commutative Algebra, Symposia Mathematica XXXIV, Cambridge University Press, Cambridge, MA, 1993, pp. 216-256.
    • (1993) Symposia Mathematica , vol.34 , pp. 216-256
    • Giusti, M.1    Heintz, J.2
  • 31
    • 84955611783 scopus 로고
    • When polynomial equation systems can be solved fast?
    • G. Cohen, M. Giusti, T. Mora (Eds.), Proc. AAECC-11, Springer, Berlin
    • M. Giusti, J. Heintz, J.E. Morais, L.M. Pardo, When polynomial equation systems can be solved fast? in: G. Cohen, M. Giusti, T. Mora (Eds.), Proc. AAECC-11, Lecture Notes in Computer Science, Vol. 948, Springer, Berlin, 1995, pp. 205-231.
    • (1995) Lecture Notes in Computer Science , vol.948 , pp. 205-231
    • Giusti, M.1    Heintz, J.2    Morais, J.E.3    Pardo, L.M.4
  • 32
    • 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
  • 33
    • 0035294718 scopus 로고    scopus 로고
    • A Gröbner free alternative for polynomial system solving
    • Giusti M., Lecerf G., Salvy B. A Gröbner free alternative for polynomial system solving. J. Complexity. 17(1):2001;154-211.
    • (2001) J. Complexity , vol.17 , Issue.1 , pp. 154-211
    • Giusti, M.1    Lecerf, G.2    Salvy, B.3
  • 34
    • 0011833225 scopus 로고    scopus 로고
    • Newton identities in the multivariate case: Pham systems
    • B. Buchberger et al. (Eds.), Gröbner Bases and Applications, Cambridge University Press, Cambridge, MA
    • M. Gonzalez-Lopez, L. Gonzalez-Vega, Newton identities in the multivariate case: Pham systems, in: B. Buchberger, et al. (Eds.), Gröbner Bases and Applications, London Math. SLNS 251, Cambridge University Press, Cambridge, MA, 1998, pp. 351-366.
    • (1998) London Math. SLNS , vol.251 , pp. 351-366
    • Gonzalez-Lopez, M.1    Gonzalez-Vega, L.2
  • 35
    • 2042530449 scopus 로고    scopus 로고
    • A special quantifier elimination algorithm for Pham systems
    • C. Detzell et al. Real algebraic geometry and ordered structures. Providence, RI: AMS
    • Gonzalez-Vega L. A special quantifier elimination algorithm for Pham systems. Detzell et al. C. Real algebraic geometry and ordered structures. Contemporary Mathematics. Vol. 253:1998;115-366 AMS, Providence, RI.
    • (1998) Contemporary Mathematics , vol.253 , pp. 115-366
    • Gonzalez-Vega, L.1
  • 36
    • 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(3):1983;239-277.
    • (1983) Theoret. Comput. Sci. , vol.24 , Issue.3 , pp. 239-277
    • Heintz, J.1
  • 37
    • 0001365155 scopus 로고
    • On the computational complexity of polynomials and bilinear mappings. A survey
    • L. Huguet, A. Poli (Eds.), Proc. AAECC-5, Menorca, Spain, June, Springer, Berlin
    • J. Heintz, On the computational complexity of polynomials and bilinear mappings. A survey, in: L. Huguet, A. Poli (Eds.), Proc. AAECC-5, Menorca, Spain, June 1987, Lecture Notes in Computer Science, Vol. 356, Springer, Berlin, 1989, pp. 269-300.
    • (1987) Lecture Notes in Computer Science , vol.356 , pp. 269-300
    • Heintz, J.1
  • 39
    • 0001787593 scopus 로고    scopus 로고
    • Deformation techniques for efficient polynomial equation solving
    • Heintz J., Krick T., Puddu S., Sabia J., Waissbein A. Deformation techniques for efficient polynomial equation solving. J. Complexity. 16(1):2000;70-109.
    • (2000) J. Complexity , vol.16 , Issue.1 , pp. 70-109
    • Heintz, J.1    Krick, T.2    Puddu, S.3    Sabia, J.4    Waissbein, A.5
  • 40
    • 0035060036 scopus 로고    scopus 로고
    • On the time-space complexity of geometric elimination procedures
    • Heintz J., Matera G., Waissbein A. On the time-space complexity of geometric elimination procedures. Appl. Algebra Engrg. Comm. Comp. 11(4):2001;239-296.
    • (2001) Appl. Algebra Engrg. Comm. Comp. , vol.11 , Issue.4 , pp. 239-296
    • Heintz, J.1    Matera, G.2    Waissbein, A.3
  • 41
    • 0003304963 scopus 로고
    • Geometric Theory of Semilinear Parabolic Equations
    • Berlin, Heideilberg: Springer
    • Henry D. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics. Vol. 840:1981;Springer, Berlin, Heideilberg.
    • (1981) Lecture Notes in Mathematics , vol.840
    • Henry, D.1
  • 42
    • 84968520678 scopus 로고
    • A polyhedral method for solving sparse polynomial systems
    • Huber B., Sturmfels B. A polyhedral method for solving sparse polynomial systems. Math. Comput. 64(112):1995;1541-1555.
    • (1995) Math. Comput. , vol.64 , Issue.112 , pp. 1541-1555
    • Huber, B.1    Sturmfels, B.2
  • 43
    • 0000697864 scopus 로고    scopus 로고
    • A computational method for Diophantine approximation
    • L. González-Vega, & T. Recio. Basel: Birkhäuser
    • Algorithms in Algebraic Geometry and Applications. Krick T., Pardo L.M. A computational method for Diophantine approximation. González-Vega L., Recio T. Algorithms in Algebraic Geometry and Applications. Progress in Mathematics. Vol. 143:1996;193-254 Birkhäuser, Basel.
    • (1996) Progress in Mathematics , vol.143 , pp. 193-254
    • Krick, T.1    Pardo, L.M.2
  • 44
    • 84941839155 scopus 로고
    • Grundzüge einer arithmetischen Theorie der algebraischen Grössen
    • Kronecker L. Grundzüge einer arithmetischen Theorie der algebraischen Grössen. J. Reine Angew. Math. 92:1882;1-122.
    • (1882) J. Reine Angew. Math. , vol.92 , pp. 1-122
    • Kronecker, L.1
  • 46
    • 0029704143 scopus 로고    scopus 로고
    • Cauchy index computation
    • Lickteig T., Roy M.-F. Cauchy index computation. Calcolo. 33:1997;337-351.
    • (1997) Calcolo , vol.33 , pp. 337-351
    • Lickteig, T.1    Roy, M.-F.2
  • 49
    • 0000372572 scopus 로고
    • An introduction to the tangent cone algorithm
    • C. Hoffmann (Ed.), Issues in robotics and non-linear geometry, JAI Press, Greenwich CT
    • T. Mora, G. Pfister, C. Traverso, An introduction to the tangent cone algorithm, in: C. Hoffmann (Ed.), Issues in robotics and non-linear geometry, Vol. 6 of Advances in Computing Research, JAI Press, Greenwich CT, 1992, pp. 199-270.
    • (1992) Advances in Computing Research , vol.6 , pp. 199-270
    • Mora, T.1    Pfister, G.2    Traverso, C.3
  • 50
    • 0001385710 scopus 로고    scopus 로고
    • Solving special polynomial systems by using structural matrices and algebraic residues
    • F. Cucker, M. Shub (Eds.), Springer, Berlin, Heidelberg, New York
    • B. Mourrain, V. Pan, Solving special polynomial systems by using structural matrices and algebraic residues, in: F. Cucker, M. Shub (Eds.), Proc. FOCM'97, Springer, Berlin, Heidelberg, New York, 1997, pp. 287-304.
    • (1997) Proc. FOCM'97 , pp. 287-304
    • Mourrain, B.1    Pan, V.2
  • 51
    • 0000692349 scopus 로고    scopus 로고
    • Multivariate polynomials, duality and structured matrices
    • Mourrain B., Pan V. Multivariate polynomials, duality and structured matrices. J. Complexity. 16(1):2000;110-180.
    • (2000) J. Complexity , vol.16 , Issue.1 , pp. 110-180
    • Mourrain, B.1    Pan, V.2
  • 52
    • 84955604612 scopus 로고
    • How lower and upper complexity bounds meet in elimination theory
    • G. Cohen, M. Giusti, T. Mora (Eds.), Proc. AAECC-11, Springer, Berlin, Heidelberg, New York
    • L.M. Pardo, How lower and upper complexity bounds meet in elimination theory, in: G. Cohen, M. Giusti, T. Mora (Eds.), Proc. AAECC-11, Lecture Notes in Computer Science, Vol. 948, Springer, Berlin, Heidelberg, New York, 1995, pp. 33-69.
    • (1995) Lecture Notes in Computer Science , vol.948 , pp. 33-69
    • Pardo, L.M.1
  • 53
    • 2042437053 scopus 로고    scopus 로고
    • Deformation techniques to solve generalized Pham systems
    • this Vol.
    • L.M. Pardo, J. San Marti;́n, Deformation techniques to solve generalized Pham systems, Theoret. Comput. Sci., this Vol. (2004).
    • (2004) Theoret. Comput. Sci.
    • Pardo, L.M.1    San Marti2    ́n, J.3
  • 54
    • 0001572510 scopus 로고
    • On the computational complexity and geometry of the first order theory of the reals I, II, III
    • Renegar J. On the computational complexity and geometry of the first order theory of the reals I, II, III. J. Symbolic Comput. 13(3):1992;255-352.
    • (1992) J. Symbolic Comput. , vol.13 , Issue.3 , pp. 255-352
    • Renegar, J.1
  • 56
    • 0033131339 scopus 로고    scopus 로고
    • Solving zero-dimensional systems throught rational univariate representation
    • Rouillier F. Solving zero-dimensional systems throught rational univariate representation. Appl. Algebra Engrg. Comm. Comp. 9(5):1997;433-461.
    • (1997) Appl. Algebra Engrg. Comm. Comp. , vol.9 , Issue.5 , pp. 433-461
    • Rouillier, F.1
  • 58
    • 0037332265 scopus 로고    scopus 로고
    • Computing parametric geometric resolutions
    • Schost E. Computing parametric geometric resolutions. Appl. Algebra Engrg. Comm. Comp. 13:2003;349-393.
    • (2003) Appl. Algebra Engrg. Comm. Comp. , vol.13 , pp. 349-393
    • Schost, E.1
  • 59
    • 84976779342 scopus 로고
    • Fast probabilistic algorithms for verification of polynomial identities
    • Schwartz J. Fast probabilistic algorithms for verification of polynomial identities. J. ACM. 27(4):1980;701-717.
    • (1980) J. ACM , vol.27 , Issue.4 , pp. 701-717
    • Schwartz, J.1
  • 61
    • 0001052336 scopus 로고
    • Algebraic complexity theory
    • van Leeuwen, J. (Ed.), Elsevier, Amsterdam, (Chapter 11)
    • V. Strassen, Algebraic complexity theory, in: van Leeuwen, J. (Ed.), Handbook of Theoretical Computer Science, Elsevier, Amsterdam, 1990, pp. 634-671 (Chapter 11).
    • (1990) Handbook of Theoretical Computer Science , pp. 634-671
    • Strassen, V.1
  • 63
    • 84957621752 scopus 로고
    • Parallel arithmetic computations: A survey
    • B.R.J. Gruska, J. Wiedermann (Eds.), Proc. 12th Symp. MFCS, Bratislava, Czechoslovakia, 1986, Springer, Berlin
    • J. von zur Gathen, Parallel arithmetic computations: a survey, in: B.R.J. Gruska, J. Wiedermann (Eds.), Proc. 12th Symp. MFCS, Bratislava, Czechoslovakia, 1986, Lecture Notes in Computer Science, Vol. 233, Springer, Berlin, 1986, pp. 93-112.
    • (1986) Lecture Notes in Computer Science , vol.233 , pp. 93-112
    • Von zur Gathen, J.1
  • 64
    • 0004055955 scopus 로고
    • New York: Dover Publications Inc.
    • Walker R. Algebraic Curves. 1950;Dover Publications Inc. New York.
    • (1950) Algebraic Curves
    • Walker, R.1
  • 65
    • 0038936302 scopus 로고    scopus 로고
    • On the complexity of rational Puiseux expansions
    • Walsh P. On the complexity of rational Puiseux expansions. Pacific J. Math. 188(2):1999;369-387.
    • (1999) Pacific J. Math. , vol.188 , Issue.2 , pp. 369-387
    • Walsh, P.1
  • 66
    • 0034400787 scopus 로고    scopus 로고
    • A polynomial-time complexity bound for the computation of the singular part of a Puiseux expansion of an algebraic, function
    • Walsh P. A polynomial-time complexity bound for the computation of the singular part of a Puiseux expansion of an algebraic, function. Math. Comput. 69(231):2000;1167-1182.
    • (2000) Math. Comput. , vol.69 , Issue.231 , pp. 1167-1182
    • Walsh, P.1
  • 67
    • 0004105164 scopus 로고
    • Berlin, Heidelberg, New York: Springer
    • Zariski O. Algebraic Surfaces. 1995;Springer, Berlin, Heidelberg, New York.
    • (1995) Algebraic Surfaces
    • Zariski, O.1
  • 68
    • 84925463455 scopus 로고
    • Probabilistic algorithms for sparse polynomials
    • Proc. EUROSAM'79, Marseille 1979, Springer, Berlin
    • R. Zippel, Probabilistic algorithms for sparse polynomials, in: Proc. EUROSAM'79, Marseille 1979, Lecture Notes in Computer Science, Vol. 72, Springer, Berlin, 1979, pp. 216-226.
    • (1979) Lecture Notes in Computer Science , vol.72 , pp. 216-226
    • Zippel, R.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.