메뉴 건너뛰기




Volumn 30, Issue 3, 2003, Pages 379-414

Counting Real Connected Components of Trinomial Curve Intersections and m-Nomial Hypersurfaces

Author keywords

[No Author keywords available]

Indexed keywords


EID: 0142054849     PISSN: 01795376     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00454-003-2834-8     Document Type: Article
Times cited : (37)

References (50)
  • 1
    • 0033436565 scopus 로고    scopus 로고
    • On Bounding the Betti Numbers and Computing the Euler Characteristic of Semi-Algebraic Sets
    • [Ba]
    • [Ba] Basu, Saugata, On Bounding the Betti Numbers and Computing the Euler Characteristic of Semi-Algebraic Sets, Discrete Comput. Geom. 22 (1999), 1-18.
    • (1999) Discrete Comput. Geom. , vol.22 , pp. 1-18
    • Basu, S.1
  • 2
    • 0039263520 scopus 로고
    • On the Number of Additions to Compute Specific Polynomials
    • [BC]
    • [BC] Borodin, Allan and Cook, Stephen A., On the Number of Additions to Compute Specific Polynomials, SIAM J. Comput. 5(1) (1976), 146-157.
    • (1976) SIAM J. Comput. , vol.5 , Issue.1 , pp. 146-157
    • Borodin, A.1    Cook, S.A.2
  • 4
    • 51249174961 scopus 로고
    • Bounding the Number of Connected Components of a Real Algebraic Set
    • [BLR]
    • [BLR] Benedetti, Riccardo, Loeser, François, and Risler, Jean-Jacques, Bounding the Number of Connected Components of a Real Algebraic Set, Discrete Comput. Geom. 6 (1991) 191-209.
    • (1991) Discrete Comput. Geom. , vol.6 , pp. 191-209
    • Benedetti, R.1    Loeser, F.2    Risler, J.-J.3
  • 7
    • 0004188587 scopus 로고
    • [BZ] Grundlehren der mathematischen Wissenschaften 285, Springer-Verlag, Berlin
    • [BZ] Burago, Yu. D., and Zalgaller, V. A., Geometric Inequalities, Grundlehren der mathematischen Wissenschaften 285, Springer-Verlag, Berlin, 1988.
    • (1988) Geometric Inequalities
    • Burago, Y.D.1    Zalgaller, V.A.2
  • 10
    • 0004233493 scopus 로고
    • [Fu1] Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 2, Springer-Verlag, Berlin
    • [Fu1] Fulton, William, Intersection Theory, 1st edn., Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 2, Springer-Verlag, Berlin, 1984.
    • (1984) Intersection Theory, 1st Edn.
    • Fulton, W.1
  • 11
    • 0003435678 scopus 로고
    • [Fu2] Annals of Mathematics Studies, no. 131, Princeton University Press, Princeton, NJ
    • [Fu2] Fulton, William, Introduction to Toric Varieties, Annals of Mathematics Studies, no. 131, Princeton University Press, Princeton, NJ, 1993.
    • (1993) Introduction to Toric Varieties
    • Fulton, W.1
  • 13
    • 0041320344 scopus 로고
    • Lower Bounds in the Algebraic Complexity of Computations, the Theory of the Complexity of Computations, I
    • [Gr1]
    • [Gr1] Grigoriev, Dima Yu., Lower Bounds in the Algebraic Complexity of Computations, The Theory of the Complexity of Computations, I, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 118 (1982), 25-82, 214.
    • (1982) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) , vol.118 , pp. 25-82
    • Grigoriev, D.Y.1
  • 14
    • 33646627778 scopus 로고    scopus 로고
    • [Gr2] personal communication, to J. Maurice Rojas, at a conference on Model Theoretical Algebra, Edinburgh, Scotland, Sept. 8
    • [Gr2] Grigoriev, Dima Yu., personal communication, to J. Maurice Rojas, at a conference on Model Theoretical Algebra, Edinburgh, Scotland, Sept. 8, 2000.
    • (2000)
    • Grigoriev, D.Y.1
  • 15
    • 33646632316 scopus 로고    scopus 로고
    • A Simple Counter-Example to Kushnirenko's Conjecture
    • [Ha]
    • [Ha] Haas, Bertrand, A Simple Counter-Example to Kushnirenko's Conjecture, Beiträge Algebra Geom., 43(1) (2002), 1-8.
    • (2002) Beiträge Algebra Geom. , vol.43 , Issue.1 , pp. 1-8
    • Haas, B.1
  • 16
    • 0004188206 scopus 로고
    • [Hi] corrected reprint of the 1976 original, Graduate Texts in Mathematics 33, Springer-Verlag, New York
    • [Hi] Hirsch, Morris, Differential Topology, corrected reprint of the 1976 original, Graduate Texts in Mathematics 33, Springer-Verlag, New York, 1994.
    • (1994) Differential Topology
    • Hirsch, M.1
  • 17
    • 0000141751 scopus 로고    scopus 로고
    • Multivariate Descartes' Rule
    • [IR]
    • [IR] Itenberg, Ilia, and Roy, Marie-Françoise, Multivariate Descartes' Rule, Beiträge Algebra Geom. 37(2) (1996), 337-346.
    • (1996) Beiträge Algebra Geom. , vol.37 , Issue.2 , pp. 337-346
    • Itenberg, I.1    Roy, M.-F.2
  • 18
    • 0013520720 scopus 로고
    • On Zeros of Exponential Sums
    • [Ka]
    • [Ka] Kazarnovskiľ, B. Ja., On Zeros of Exponential Sums, Soviet Math. Dokl., 23(2) (1981), 347-351.
    • (1981) Soviet Math. Dokl. , vol.23 , Issue.2 , pp. 347-351
    • Kazarnovskiľ, B.J.1
  • 19
    • 0003376293 scopus 로고
    • On a Class of Systems of Transcendental Equations
    • [Kh1]
    • [Kh1] Khovanski, Askold Georgievich, On a Class of Systems of Transcendental Equations, Dokl. Akad. Nauk SSSR 255(4) (1980), 804-807; English transl, in Soviet Math. Dokl. 22(3) (1980).
    • (1980) Dokl. Akad. Nauk SSSR , vol.255 , Issue.4 , pp. 804-807
    • Khovanski, A.G.1
  • 20
    • 84922441149 scopus 로고
    • [Kh1] Khovanski, Askold Georgievich, On a Class of Systems of Transcendental Equations, Dokl. Akad. Nauk SSSR 255(4) (1980), 804-807; English transl, in Soviet Math. Dokl. 22(3) (1980).
    • (1980) Soviet Math. Dokl. , vol.22 , Issue.3
  • 21
    • 0004015202 scopus 로고
    • [Kh2] AMS Press, Providence, RI
    • [Kh2] Khovanski, Askold Georgievich, Fewnomials, AMS Press, Providence, RI, 1991.
    • (1991) Fewnomials
    • Khovanski, A.G.1
  • 22
    • 33646602659 scopus 로고    scopus 로고
    • [Kh3] personal communication, via e-mails to J. Maurice Rojas, August 23-September 1
    • [Kh3] Khovanski, Askold Georgievich, personal communication, via e-mails to J. Maurice Rojas, August 23-September 1, 2002.
    • (2002)
    • Khovanski, A.G.1
  • 23
    • 0040151555 scopus 로고    scopus 로고
    • Multivariate Descartes Rule of Signs and Sturmfels' Challenge Problem
    • [LR]
    • [LR] Lagarias, Jeffrey C., and Richardson, Thomas J., Multivariate Descartes Rule of Signs and Sturmfels' Challenge Problem, Math. Intelligencer 19(3) (1997), 9-15.
    • (1997) Math. Intelligencer , vol.19 , Issue.3 , pp. 9-15
    • Lagarias, J.C.1    Richardson, T.J.2
  • 24
    • 0040843823 scopus 로고    scopus 로고
    • On Multivariate Descartes' Rule - A Counter-Example
    • [LW]
    • [LW] Li, Tien-Yien, and Wang, Xiaoshen, On Multivariate Descartes' Rule - A Counter-Example, Beiträge Algebra Geom. 39(1) (1998), 1-5.
    • (1998) Beiträge Algebra Geom. , vol.39 , Issue.1 , pp. 1-5
    • Li, T.-Y.1    Wang, X.2
  • 25
    • 0004190360 scopus 로고
    • [Mi1] based on lecture notes by M. Spivak and R. Wells, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, NJ
    • [Mi1] Milnor, John, Morse Theory, based on lecture notes by M. Spivak and R. Wells, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, NJ, 1963.
    • (1963) Morse Theory
    • Milnor, J.1
  • 26
    • 84966260816 scopus 로고
    • On the Betti Numbers of Real Varieties
    • [Mi2]
    • [Mi2] Milnor, John, 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
  • 28
    • 33646603198 scopus 로고
    • [Ne] edited by G. F. Salvemini, Marc-Michel Bosquet, Lausanne and Geneva
    • [Ne] Newton, Sir Isaac, Opuscula Mathematica, Philosophies et Philologica, edited by G. F. Salvemini, 3 volumes, Marc-Michel Bosquet, Lausanne and Geneva, 1744.
    • (1744) Opuscula Mathematica, Philosophies et Philologica , vol.3
    • Newton, S.I.1
  • 29
    • 0001435982 scopus 로고
    • On the Topology of Real Algebraic Hypersurfaces
    • [OP]
    • [OP] Oleinik, O., and Petrovsky, I., On the Topology of Real Algebraic Hypersurfaces, Izv. Akad. Nauk SSSR Ser. Mat. 13 (1949), 389-402; English transl., Amer. Math. Soc. Transl. (1) 7 (1962), 399-417.
    • (1949) Izv. Akad. Nauk SSSR Ser. Mat. , vol.13 , pp. 389-402
    • Oleinik, O.1    Petrovsky, I.2
  • 30
    • 0345100090 scopus 로고
    • [OP] Oleinik, O., and Petrovsky, I., On the Topology of Real Algebraic Hypersurfaces, Izv. Akad. Nauk SSSR Ser. Mat. 13 (1949), 389-402; English transl., Amer. Math. Soc. Transl. (1) 7 (1962), 399-417.
    • (1962) Amer. Math. Soc. Transl. , vol.7 , Issue.1 , pp. 399-417
  • 31
    • 0041869417 scopus 로고    scopus 로고
    • Simple Non-Rational Convex Polytopes via Symplectic Geometry
    • [Pr]
    • [Pr] Prato, Elisa, Simple Non-Rational Convex Polytopes via Symplectic Geometry, Topology 40(5) (2001), 961-975.
    • (2001) Topology , vol.40 , Issue.5 , pp. 961-975
    • Prato, E.1
  • 32
    • 0024646216 scopus 로고
    • On the Worst Case Arithmetic Complexity of Approximating Zeros of Systems of Polynomials
    • [Re]
    • [Re] Renegar, Jim, On the Worst Case Arithmetic Complexity of Approximating Zeros of Systems of Polynomials, SIAM J. Comput. 18(2) (1989), 350-370.
    • (1989) SIAM J. Comput. , vol.18 , Issue.2 , pp. 350-370
    • Renegar, J.1
  • 33
    • 0022010891 scopus 로고
    • Additive Complexity and Zeros of Real Polynomials
    • [Ri]
    • [Ri] Risler, Jean-Jacques, Additive Complexity and Zeros of Real Polynomials, SIAM J. Comput. 14(1) (1985), 178-183.
    • (1985) SIAM J. Comput. , vol.14 , Issue.1 , pp. 178-183
    • Risler, J.-J.1
  • 34
    • 0033522024 scopus 로고    scopus 로고
    • Toric Intersection Theory for Affine Root Counting
    • [Ro1]
    • [Ro1] Rojas, J. Maurice, Toric Intersection Theory for Affine Root Counting, J. Pure Appl. Algebra, 136(1) (1999), 67-100.
    • (1999) J. Pure Appl. Algebra , vol.136 , Issue.1 , pp. 67-100
    • Rojas, J.M.1
  • 35
    • 0005309206 scopus 로고    scopus 로고
    • Some Speed-Ups and Speed Limits for Real Algebraic Geometry
    • [Ro2] FoCM 1999 special issue
    • [Ro2] Rojas, J. Maurice, Some Speed-Ups and Speed Limits for Real Algebraic Geometry, J. Complexity (FoCM 1999 special issue) 16(3) (2000), 552-571.
    • (2000) J. Complexity , vol.16 , Issue.3 , pp. 552-571
    • Rojas, J.M.1
  • 37
    • 84922425707 scopus 로고    scopus 로고
    • Arithmetic Multivariate Descartes' Rule
    • [Ro4] to appear
    • [Ro4] Rojas, J. Maurice, Arithmetic Multivariate Descartes' Rule, Amer. J. Math., to appear.
    • Amer. J. Math.
    • Rojas, J.M.1
  • 40
    • 0004287752 scopus 로고
    • [Sh] Springer Study Edition, Springer-Verlag, New York
    • [Sh] Shafarevich, Igor R., Basic Algebraic Geometry, Springer Study Edition, Springer-Verlag, New York, 1977.
    • (1977) Basic Algebraic Geometry
    • Shafarevich, I.R.1
  • 41
    • 0004266718 scopus 로고
    • [SL] translated from the French and Latin (with a facsimile of Descartes' 1637 French edition), Dover, New York
    • [SL] Smith, David Eugene, and Latham, Marcia L., The Geometry of René Descartes, translated from the French and Latin (with a facsimile of Descartes' 1637 French edition), Dover, New York, 1954.
    • (1954) The Geometry of René Descartes
    • Smith, D.E.1    Latham, M.L.2
  • 42
    • 22244441450 scopus 로고
    • Topology and Mechanics I
    • [Sm]
    • [Sm] Smale, Steve, Topology and Mechanics I, Invent. Math. 10 (1970), 305-331.
    • (1970) Invent. Math. , vol.10 , pp. 305-331
    • Smale, S.1
  • 44
    • 0008599709 scopus 로고
    • On the Number of Real Roots of a Sparse Polynomial System
    • [St1] Fields Inst. Commun. 3, American Mathematical Society, Providence, RI
    • [St1] Sturmfels, Bernd, On the Number of Real Roots of a Sparse Polynomial System, Hamiltonian and Gradient Flows, Algorithms and Control, pp. 137-143, Fields Inst. Commun. 3, American Mathematical Society, Providence, RI, 1994.
    • (1994) Hamiltonian and Gradient Flows, Algorithms and Control , pp. 137-143
    • Sturmfels, B.1
  • 45
    • 0032264839 scopus 로고    scopus 로고
    • Polynomial Equations and Convex Polytopes
    • [St2]
    • [St2] Sturmfels, Bernd, Polynomial Equations and Convex Polytopes, Amer. Math. Monthly 105(10) (1998), 907-922.
    • (1998) Amer. Math. Monthly , vol.105 , Issue.10 , pp. 907-922
    • Sturmfels, B.1
  • 46
    • 0002586126 scopus 로고
    • Sur l'homologie des variétés algébriques réelles
    • [Th] S. Cairns (ed.), Princeton University Press, Princeton, NJ
    • [Th] Thom, René, Sur l'homologie des variétés algébriques réelles, in S. Cairns (ed.), Differential and Combinatorial Topology, Princeton University Press, Princeton, NJ, 1965.
    • (1965) Differential and Combinatorial Topology
    • Thom, R.1
  • 47
    • 0001656792 scopus 로고    scopus 로고
    • Algorithm 795: PHCpack: A General-Purpose Solver for Polynomial Systems by Homotopy Continuation
    • [Ve]
    • [Ve] Verscheide, Jan, Algorithm 795: PHCpack: A General-Purpose Solver for Polynomial Systems by Homotopy Continuation, ACM Trans. Math. Software 25(2): (1999), 251-276.
    • (1999) ACM Trans. Math. Software , vol.25 , Issue.2 , pp. 251-276
    • Verscheide, J.1
  • 48
    • 0000403820 scopus 로고
    • Gluing of Plane Real Algebraic Curves and Constructions of Curves of Degrees 6 and 7
    • [Vi] Lecture Notes in Mathematics 1060, Springer-Verlag, Berlin
    • [Vi] Viro, Oleg Ya., Gluing of Plane Real Algebraic Curves and Constructions of Curves of Degrees 6 and 7, Topology (Leningrad, 1982), pp. 187-200, Lecture Notes in Mathematics 1060, Springer-Verlag, Berlin, 1984.
    • (1984) Topology (Leningrad, 1982) , pp. 187-200
    • Viro, O.Ya.1
  • 50
    • 0033145513 scopus 로고    scopus 로고
    • Betti Numbers of Semi-Pfaffian Sets, (Effective Methods in Algebraic Geometry (MEGA '98, Saint-Malo, 1998))
    • [Ze]
    • [Ze] Zell, Thierry, Betti Numbers of Semi-Pfaffian Sets, (Effective Methods in Algebraic Geometry (MEGA '98, Saint-Malo, 1998)), J. Pure Appl. Algebra 139(1-3) (1999), 323-338.
    • (1999) J. Pure Appl. Algebra , vol.139 , Issue.1-3 , pp. 323-338
    • Zell, T.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.