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Volumn 10, Issue 3, 1999, Pages 191-220

Algebraic geometry and computer vision: Polynomial systems, real and complex roots

(1)  Petitjean, S a  

a INRIA   (France)

Author keywords

[No Author keywords available]

Indexed keywords

LINEAR ALGEBRA; POLYNOMIALS;

EID: 0032689464     PISSN: 09249907     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1008348724781     Document Type: Article
Times cited : (20)

References (118)
  • 3
    • 0345358213 scopus 로고
    • Geometric reasoning with logic and algebra
    • Geometric Reasoning, D. Kapur and J. Mundy (Eds), MIT Press
    • D. Arnon, "Geometric reasoning with logic and algebra," in Geometric Reasoning, D. Kapur and J. Mundy (Eds), MIT Press, 1989, pp. 37-60. Proceedings of the International Workshop on Geometric Reasoning.
    • (1989) Proceedings of the International Workshop on Geometric Reasoning , pp. 37-60
    • Arnon, D.1
  • 5
    • 77956969818 scopus 로고
    • An introduction to first-order logic
    • J. Barwise (Ed.), North Holland
    • J. Barwise, "An introduction to first-order logic," in Handbook of Mathematical Logic, J. Barwise (Ed.), North Holland, 1977, pp. 5-46.
    • (1977) Handbook of Mathematical Logic , pp. 5-46
    • Barwise, J.1
  • 6
    • 0002128307 scopus 로고
    • What can becomputed in algebraic geometry?
    • Computational Algebraic Geometry and Commutative Algebra Cortona, Cambridge University Press
    • D. Bayer and D. Mumford, "What can becomputed in algebraic geometry?" in Computational Algebraic Geometry and Commutative Algebra (Cortona, 1991), Sympos. Math., XXXIV, Cambridge University Press, 1993, pp. 1-48.
    • (1991) Sympos. Math., XXXIV , pp. 1-48
    • Bayer, D.1    Mumford, D.2
  • 8
  • 11
    • 34250406508 scopus 로고
    • The number of rootsof a system of equations
    • D. N. Bernstein, "The number of rootsof a system of equations," Functional Analysis and Applications, Vol. 9, No. 2, pp. 183-185, 1975.
    • (1975) Functional Analysis and Applications , vol.9 , Issue.2 , pp. 183-185
    • Bernstein, D.N.1
  • 14
    • 0002803133 scopus 로고
    • Gröbner bases: An algorithmic method in polynomial ideal theory
    • N.K. Bose (Ed.), Reidel: Dordrecht-Boston-Lancaster
    • B. Buchberger, "Gröbner bases: An algorithmic method in polynomial ideal theory," in Multidimensional Systems Theory, N.K. Bose (Ed.), Reidel: Dordrecht-Boston-Lancaster, 1985, pp. 184-232.
    • (1985) Multidimensional Systems Theory , pp. 184-232
    • Buchberger, B.1
  • 15
    • 0008660871 scopus 로고
    • Applications of Gröbner bases in non-linear computational geometry
    • D. Kapur and J. Mundy (Eds), The MIT Press
    • B. Buchberger, "Applications of Gröbner bases in non-linear computational geometry," in Geometric Reasoning, D. Kapur and J. Mundy (Eds), The MIT Press, 1989, pp. 415-447.
    • (1989) Geometric Reasoning , pp. 415-447
    • Buchberger, B.1
  • 19
    • 0026191608 scopus 로고
    • A singly exponential stratification scheme for real semi-algebraic varieties and its applications
    • B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir, "A singly exponential stratification scheme for real semi-algebraic varieties and its applications," Theoretical Computer Science, Vol. 84, pp. 77-105, 1991.
    • (1991) Theoretical Computer Science , vol.84 , pp. 77-105
    • Chazelle, B.1    Edelsbrunner, H.2    Guibas, L.3    Sharir, M.4
  • 21
    • 0001560099 scopus 로고
    • Thorn's lemma, the coding of real algebraic numbers and the computation of the topology of semi-algebrac sets
    • M. Coste and M.-F. Roy, "Thorn's lemma, the coding of real algebraic numbers and the computation of the topology of semi-algebrac sets," Journal of Symbolic Computation, Vol. 5, pp. 121-129, 1988.
    • (1988) Journal of Symbolic Computation , vol.5 , pp. 121-129
    • Coste, M.1    Roy, M.-F.2
  • 24
    • 84960589566 scopus 로고
    • The eliminant of three quantics in two independent variables
    • A.L. Dixon, "The eliminant of three quantics in two independent variables," Proceedings of the London Mathematical Society, Vol. 6, pp. 49-69, 1908.
    • (1908) Proceedings of the London Mathematical Society , vol.6 , pp. 49-69
    • Dixon, A.L.1
  • 25
    • 0000348667 scopus 로고
    • Efficient incremental algorithms for the sparse resultant and the mixed volume
    • I. Emiris and J. Canny, "Efficient incremental algorithms for the sparse resultant and the mixed volume," Journal of Symbolic Computation, Vol. 20, No. 2, pp. 117-149, 1995.
    • (1995) Journal of Symbolic Computation , vol.20 , Issue.2 , pp. 117-149
    • Emiris, I.1    Canny, J.2
  • 27
    • 25944479355 scopus 로고
    • Efficient computation of zero-dimensional Gröbner bases by change of ordering
    • J.-C. Faugère, P. Gianni, D. Lazard, and T. Mora, "Efficient computation of zero-dimensional Gröbner bases by change of ordering," Journal of Symbolic Computation, Vol. 16, No. 4, pp. 329-344, 1993.
    • (1993) Journal of Symbolic Computation , vol.16 , Issue.4 , pp. 329-344
    • Faugère, J.-C.1    Gianni, P.2    Lazard, D.3    Mora, T.4
  • 31
    • 25544447056 scopus 로고
    • Some examples of problem solving by using the symbolic viewpoint when dealing with polynomial systems of equations
    • J. Fleischer, J. Grabmeier, F.W. Hehl, and W. Küchlin (Eds), World Scientific Publishing
    • L. González-Vega, "Some examples of problem solving by using the symbolic viewpoint when dealing with polynomial systems of equations," in Computer Algebra in Science and Engineering, J. Fleischer, J. Grabmeier, F.W. Hehl, and W. Küchlin (Eds), World Scientific Publishing, 1995, pp. 102-116.
    • (1995) Computer Algebra in Science and Engineering , pp. 102-116
    • González-Vega, L.1
  • 34
    • 25544450035 scopus 로고    scopus 로고
    • About the polynomial system solve facility of Axiom, Macsyma, Maple, Mathematica, MuPAD, and Reduce
    • M. Wester (Ed.), John Wiley, to appear
    • H.-G. Gräbe, "About the polynomial system solve facility of Axiom, Macsyma, Maple, Mathematica, MuPAD, and Reduce," in Computer Algebra Systems: A Practical Guide, M. Wester (Ed.), John Wiley, 1998, to appear.
    • (1998) Computer Algebra Systems: A Practical Guide
    • Gräbe, H.-G.1
  • 37
    • 0344495756 scopus 로고
    • Gröbner bases
    • G. Heizer, "Gröbner bases," Mathematical Journal, Vol. 5, No. 1, pp. 67-73, 1995.
    • (1995) Mathematical Journal , vol.5 , Issue.1 , pp. 67-73
    • Heizer, G.1
  • 39
    • 0004249123 scopus 로고
    • Cambridge Mathematica Library, Cambridge University Press
    • D. Hubert, Theory of Algebraic Invariants, Cambridge Mathematica Library, Cambridge University Press, 1890.
    • (1890) Theory of Algebraic Invariants
    • Hubert, D.1
  • 40
    • 0000596483 scopus 로고
    • Über die theorie der algebraischen formen
    • D. Hubert, "Über die theorie der algebraischen formen," Math. Annal en, Vol. 36, pp. 473-534, 1890.
    • (1890) Math. Annal en , vol.36 , pp. 473-534
    • Hubert, D.1
  • 42
    • 84968520678 scopus 로고
    • A polyhedral method for solving sparse polynomial systems
    • B. Huber and B. Sturmfels, "A polyhedral method for solving sparse polynomial systems," Math. Comp., Vol. 64, pp. 1541-1555, 1995.
    • (1995) Math. Comp. , vol.64 , pp. 1541-1555
    • Huber, B.1    Sturmfels, B.2
  • 46
    • 0344495749 scopus 로고
    • Cylindrical algebraic decomposition - An introduction
    • Department of Electrical Engineering, Linköping University
    • M. Jirstrand," Cylindrical algebraic decomposition - An introduction," Technical Report LiTH-ISY-R-1807, Department of Electrical Engineering, Linköping University, 1995.
    • (1995) Technical Report LiTH-ISY-R-1807
    • Jirstrand, M.1
  • 47
    • 38149144730 scopus 로고
    • Le formalisme du résultant
    • J.-P Jouanolou, "Le formalisme du résultant," Advances in Mathematics, Vol. 98, pp. 117-263, 1991.
    • (1991) Advances in Mathematics , vol.98 , pp. 117-263
    • Jouanolou, J.-P.1
  • 48
    • 42549171482 scopus 로고
    • On the average number of real roots of a random algebraic equation
    • M. Kac, "On the average number of real roots of a random algebraic equation," Bulletin of the American Mathematical Society, Vol. 49, pp. 314-320, 1943.
    • (1943) Bulletin of the American Mathematical Society , vol.49 , pp. 314-320
    • Kac, M.1
  • 49
  • 51
    • 0000441782 scopus 로고
    • Eine neue Relation zwischen den Singularitäten einer algebraischen Curve
    • F. Klein," Eine neue Relation zwischen den Singularitäten einer algebraischen Curve," Math. Annalen, Vol. 10, pp. 199-209, 1876.
    • (1876) Math. Annalen , vol.10 , pp. 199-209
    • Klein, F.1
  • 52
    • 0018777412 scopus 로고
    • The internal representation of solid shape with respect to vision
    • J.J. Koenderink and A.J. van Doom, "The internal representation of solid shape with respect to vision," Biological Cybernetics, Vol. 32, pp. 211-216, 1979.
    • (1979) Biological Cybernetics , vol.32 , pp. 211-216
    • Koenderink, J.J.1    Van Doom, A.J.2
  • 53
    • 0344064552 scopus 로고
    • On the distribution of the roots of random polynomials
    • From Topology to Computation, M. Hirsch, J. Marsden, and M. Shub (Eds), Springer-Verlag
    • E. Kostlan, "On the distribution of the roots of random polynomials," in From Topology to Computation, M. Hirsch, J. Marsden, and M. Shub (Eds), Springer-Verlag, 1991. Proceedings of the Smalefest.
    • (1991) Proceedings of the Smalefest
    • Kostlan, E.1
  • 54
    • 0002548511 scopus 로고
    • Résolution des systèmes d' équations algébriques
    • D. Lazard, "Résolution des systèmes d' équations algébriques," Theoretical Computer Science, Vol. 15, pp. 77-110, 1981.
    • (1981) Theoretical Computer Science , vol.15 , pp. 77-110
    • Lazard, D.1
  • 56
    • 0000534684 scopus 로고
    • Minimal resultant systems
    • G. Lyubeznik, "Minimal resultant systems," Journal of Algebra, Vol. 177, pp. 612-616, 1995.
    • (1995) Journal of Algebra , vol.177 , pp. 612-616
    • Lyubeznik, G.1
  • 59
    • 84983660640 scopus 로고
    • The implicit representation of rational parametric surfaces
    • D. Manocha and J.F. Canny, "The implicit representation of rational parametric surfaces," Journal of Symbolic Computation, Vol. 13, pp. 485-510, 1992.
    • (1992) Journal of Symbolic Computation , vol.13 , pp. 485-510
    • Manocha, D.1    Canny, J.F.2
  • 67
    • 0342583508 scopus 로고
    • The power of combining the techniques of algebraic and numerical computing: Improved approximate multipoint polynomial evaluation and improved multipole algorithms
    • V.Y. Pan, J. H. Reif, and S.R. Tate, "The power of combining the techniques of algebraic and numerical computing: Improved approximate multipoint polynomial evaluation and improved multipole algorithms," in Proceedings of FOCS'92 (IEEE Symposium on Foundations of Computer Science), 1992, pp. 703-713.
    • (1992) Proceedings of FOCS'92 (IEEE Symposium on Foundations of Computer Science) , pp. 703-713
    • Pan, V.Y.1    Reif, J.H.2    Tate, S.R.3
  • 69
    • 0030217159 scopus 로고    scopus 로고
    • The enumerative geometry of projective algebraic surfaces and the complexity of aspect graphs
    • S. Petitjean, "The enumerative geometry of projective algebraic surfaces and the complexity of aspect graphs," International Journal of Computer Vision, Vol. 19, No. 3, pp. 1-27, 1996.
    • (1996) International Journal of Computer Vision , vol.19 , Issue.3 , pp. 1-27
    • Petitjean, S.1
  • 70
    • 0344064544 scopus 로고    scopus 로고
    • Recognizing algebraic surfaces from aspects
    • J. Ponce (Ed.), Springer-Verlag, Series in Information Sciences, to appear
    • S. Petitjean, J. Rieger, and D. Forsyth, "Recognizing algebraic surfaces from aspects," in Algebraic Surfaces in Computer Vision, J. Ponce (Ed.), Springer-Verlag, 1997. Series in Information Sciences, to appear.
    • (1997) Algebraic Surfaces in Computer Vision
    • Petitjean, S.1    Rieger, J.2    Forsyth, D.3
  • 72
    • 0003583062 scopus 로고    scopus 로고
    • Ph.D. thesis, Computer Science Department, University of California at Berkeley
    • A. Rege, "A toolkit for algebra and geometry," Ph.D. thesis, Computer Science Department, University of California at Berkeley, 1996.
    • (1996) A Toolkit for Algebra and Geometry
    • Rege, A.1
  • 73
    • 0001572510 scopus 로고
    • On the computational complexity and geometry of the first-order theory of the reals I, II and III
    • J. Renegar, "On the computational complexity and geometry of the first-order theory of the reals I, II and III," Journal of Symbolic Computation, Vol. 13, No. 3, pp. 255-300, 301-328, 329-330, 1992.
    • (1992) Journal of Symbolic Computation , vol.13 , Issue.3 , pp. 255-300
    • Renegar, J.1
  • 74
    • 38249003237 scopus 로고
    • Computing view graphs of algebraic surfaces
    • J.H. Rieger, "Computing view graphs of algebraic surfaces," Journal of Symbolic Computation, Vol. 16, pp. 259-272, 1993.
    • (1993) Journal of Symbolic Computation , vol.16 , pp. 259-272
    • Rieger, J.H.1
  • 75
    • 0040676924 scopus 로고    scopus 로고
    • Notes on the complexity of exact view graph algorithms for piecewise-smooth algebraic surfaces
    • J.H. Rieger, "Notes on the complexity of exact view graph algorithms for piecewise-smooth algebraic surfaces," Discrete and Computational Geometry, Vol. 20, pp. 205-229, 1998.
    • (1998) Discrete and Computational Geometry , vol.20 , pp. 205-229
    • Rieger, J.H.1
  • 76
    • 3242783490 scopus 로고    scopus 로고
    • On the complexity and computation of view graphsof piecewise-smooth algebraic surfaces
    • J.H. Rieger, "On the complexity and computation of view graphsof piecewise-smooth algebraic surfaces," Philosophical Transactions of the Royal Society of London, Series A, Vol. 354, No. 1714, pp. 1899-1940, 1996.
    • (1996) Philosophical Transactions of the Royal Society of London, Series a , vol.354 , Issue.1714 , pp. 1899-1940
    • Rieger, J.H.1
  • 77
    • 0028517599 scopus 로고
    • A convex geometri c approach to counting the roots of a polynomial system
    • M. Rojas, "A convex geometri c approach to counting the roots of a polynomial system," Theoretical Computer Science, Vol. 133, No. 1, pp. 105-140, 1994.
    • (1994) Theoretical Computer Science , vol.133 , Issue.1 , pp. 105-140
    • Rojas, M.1
  • 78
    • 0009844162 scopus 로고    scopus 로고
    • On the average number of real roots of certain random sparse polynomial systems
    • The Mathematics of Numerical Analysis, Jim Renegar, Mike Shub, and Steve Smale (Eds.)
    • M. Rojas, "On the average number of real roots of certain random sparse polynomial systems," in The Mathematics of Numerical Analysis, Jim Renegar, Mike Shub, and Steve Smale (Eds.), volume 32 of Lectures in Applied Mathematics, American Mathematical Society, 1996, pp. 689-699.
    • (1996) Lectures in Applied Mathematics, American Mathematical Society , vol.32 , pp. 689-699
    • Rojas, M.1
  • 80
    • 0033522024 scopus 로고    scopus 로고
    • Toric intersection theory for affine root counting
    • M. Rojas, "Toric intersection theory for affine root counting," Journal of Pure and Applied Algebra, Vol. 136, No. 1, 1999.
    • (1999) Journal of Pure and Applied Algebra , vol.136 , Issue.1
    • Rojas, M.1
  • 81
    • 0030163190 scopus 로고    scopus 로고
    • Counting affine roots via pointed Newton polytopes
    • M. Rojas and X. Wang, "Counting affine roots via pointed Newton polytopes," Journal of Complexity, Vol. 12, pp. 116-133, 1996.
    • (1996) Journal of Complexity , vol.12 , pp. 116-133
    • Rojas, M.1    Wang, X.2
  • 82
    • 0345358173 scopus 로고    scopus 로고
    • Points d'inflexion sur les courbes réelles: Un travail de Félix Klein à la lumière de la théorie des singularités d'applications
    • F. Ronga, "Points d'inflexion sur les courbes réelles: un travail de Félix Klein à la lumière de la théorie des singularités d'applications," Gazette des mathématiciens, No. 74, 1997.
    • (1997) Gazette des Mathématiciens , Issue.74
    • Ronga, F.1
  • 83
    • 0001805770 scopus 로고    scopus 로고
    • The number of conic stangent to 5 given conics: The real case
    • F. Ronga, "The number of conic stangent to 5 given conics: The real case," Revista Matemática dela Universidad Complutense de Madrid, Vol. 10, No. 2, pp. 391-421, 1997.
    • (1997) Revista Matemática Dela Universidad Complutense de Madrid , vol.10 , Issue.2 , pp. 391-421
    • Ronga, F.1
  • 87
    • 0344926758 scopus 로고    scopus 로고
    • Aspect graphs of bodies of revolution with algorithms of real algebraic geometry
    • M.-F. Roy and T. van Effelterre, "Aspect graphs of bodies of revolution with algorithms of real algebraic geometry," Progress in Mathematics, 1996, Vol. 143. Proceedings of MEGA'94 (International Symposium on Effective Methods in Algebraic Geometry), Santander, Spain, April 1994.
    • (1996) Progress in Mathematics , vol.143
    • Roy, M.-F.1    Van Effelterre, T.2
  • 89
    • 0003397364 scopus 로고    scopus 로고
    • Ph.D. thesis, Utrecht University, The Netherlands
    • A. Salden, "Dynamic scale-space paradigms," Ph.D. thesis, Utrecht University, The Netherlands, 1996.
    • (1996) Dynamic Scale-space Paradigms
    • Salden, A.1
  • 93
    • 0000537756 scopus 로고
    • Complexity of Bezout's theorem II: Volumes and probabilities
    • Computational Algebraic Geometry, F. Eyssette and A. Galligo (Eds), Birkhäuser
    • M. Shub and S. Smale, "Complexity of Bezout's theorem II: Volumes and probabilities," in Computational Algebraic Geometry, F. Eyssette and A. Galligo (Eds), volume 109 of Progressin Mathematics, Birkhäuser, 1993, pp. 267-285.
    • (1993) Progressin Mathematics , vol.109 , pp. 267-285
    • Shub, M.1    Smale, S.2
  • 94
    • 0028516294 scopus 로고
    • Complexity of Bezout'stheorem V: Polynomial time
    • M. Shub and S. Smale, "Complexity of Bezout'stheorem V: Polynomial time," Theoretical Computer Science, Vol. 133, pp. 141-164, 1994.
    • (1994) Theoretical Computer Science , vol.133 , pp. 141-164
    • Shub, M.1    Smale, S.2
  • 95
    • 0000675799 scopus 로고    scopus 로고
    • Complexity of Bezout's theorem IV: Probability of success, extensions
    • M. Shub and S. Smale, "Complexity of Bezout's theorem IV: Probability of success, extensions," SIAM Journal of Numerical Analysis, Vol. 33, No. 1, pp. 128-148, 1996.
    • (1996) SIAM Journal of Numerical Analysis , vol.33 , Issue.1 , pp. 128-148
    • Shub, M.1    Smale, S.2
  • 96
    • 0000718224 scopus 로고    scopus 로고
    • Enumerative geometry for real varieties
    • Algebraic Geometry, Santa Cruz, 1995, János Kollár (Ed.), AMS
    • F. Sottile, "Enumerative geometry for real varieties," in Algebraic Geometry, Santa Cruz, 1995, János Kollár (Ed.), Proceedings and Symposia in Pure Mathematics, Vol. 61, No. 1, AMS, pp. 435-447, 1997.
    • (1997) Proceedings and Symposia in Pure Mathematics , vol.61 , Issue.1 , pp. 435-447
    • Sottile, F.1
  • 97
    • 0002925584 scopus 로고    scopus 로고
    • Enumerative geometry for the real Grassmannian of lines in projective space
    • F. Sottile, "Enumerative geometry for the real Grassmannian of lines in projective space," Duke Math. J., Vol. 87, No. 1, pp. 59-85, 1997.
    • (1997) Duke Math. J. , vol.87 , Issue.1 , pp. 59-85
    • Sottile, F.1
  • 99
    • 0344926756 scopus 로고    scopus 로고
    • An introduction to the theory of resultants
    • Texas A & M University, Institute for Scientific Computation
    • P. Stiller, "An introduction to the theory of resultants," Technical Report ISC-96-02-MATH, Texas A & M University, Institute for Scientific Computation, 1996.
    • (1996) Technical Report ISC-96-02-MATH
    • Stiller, P.1
  • 100
    • 0344495708 scopus 로고    scopus 로고
    • Sparse resultants
    • Texas A & M University, Institute for Scientific Computation
    • P. Stiller, "Sparse resultants," Technical Report ISC-96-01-MATH, Texas A & M University, Institute for Scientific Computation, 1996.
    • (1996) Technical Report ISC-96-01-MATH
    • Stiller, P.1
  • 102
    • 0000803832 scopus 로고
    • Mémoire sur la résolution des équations numériques
    • C. Sturm, "Mémoire sur la résolution des équations numériques," Inst. France Sc. Math. Phys, Vol. 6, 1835.
    • (1835) Inst. France Sc. Math. Phys , vol.6
    • Sturm, C.1
  • 103
    • 0008599709 scopus 로고
    • On the number of real roots of a sparse polynomial system
    • A. Bloch (Ed.), American Mathematical Society
    • B. Sturmfels, "On the number of real roots of a sparse polynomial system," in Hamiltonian and Gradient Flows Algorithms and Control, A. Bloch (Ed.), American Mathematical Society, 1991, Vol. 3, pp. 137-143.
    • (1991) Hamiltonian and Gradient Flows Algorithms and Control , vol.3 , pp. 137-143
    • Sturmfels, B.1
  • 105
    • 0001374975 scopus 로고
    • Multigraded resultants of Sylvester type
    • B. Sturmfels and A. Zelevinsky, "Multigraded resultants of Sylvester type," Journal of Algebra, Vol. 163, No. 1, pp. 115-127, 1994.
    • (1994) Journal of Algebra , vol.163 , Issue.1 , pp. 115-127
    • Sturmfels, B.1    Zelevinsky, A.2
  • 106
    • 0003207762 scopus 로고
    • On a general method of determining by mere inspection the derivations from two equations of any degree
    • J.J. Sylvester, "On a general method of determining by mere inspection the derivations from two equations of any degree," Philosophical Magazine, Vol. 16, pp. 132-135, 1840.
    • (1840) Philosophical Magazine , vol.16 , pp. 132-135
    • Sylvester, J.J.1
  • 110
    • 0344495706 scopus 로고
    • Approximate conversion from parametric to implicit surfaces
    • B. Wyvill and M.-P Gascuel (Eds.), Grenoble, France, April
    • L. Velho and J. Gomes, "Approximate conversion from parametric to implicit surfaces," in Proceedings of First International Workshop on Implicit Surfaces, B. Wyvill and M.-P Gascuel (Eds.), Grenoble, France, April 1995, pp. 77-96.
    • (1995) Proceedings of First International Workshop on Implicit Surfaces , pp. 77-96
    • Velho, L.1    Gomes, J.2
  • 111
    • 0008416734 scopus 로고
    • Solving parametric polynomial equations and inequalities by symbolic algorithms
    • J. Flascher, J. Grabmeier, F.W. Hehl, and W. Küchlin (Eds.), World Scientific Publishing
    • V. Weispfenning, "Solving parametric polynomial equations and inequalities by symbolic algorithms," in Computer Algebra in Science and Engineering, J. Flascher, J. Grabmeier, F.W. Hehl, and W. Küchlin (Eds.), World Scientific Publishing, 1995.
    • (1995) Computer Algebra in Science and Engineering
    • Weispfenning, V.1
  • 114
    • 0001518467 scopus 로고
    • Determinants formulas for multigraded resultants
    • J. Weyman and A. Zelevinsky, "Determinants formulas for multigraded resultants," Journal of Algebraic Geometry, Vol. 3, pp. 569-597, 1994.
    • (1994) Journal of Algebraic Geometry , vol.3 , pp. 569-597
    • Weyman, J.1    Zelevinsky, A.2
  • 115
    • 0038256188 scopus 로고
    • On the decision problem and the mechanization of theorem-proving in elementary geometry
    • W. Bledsoe and D. Loveland (Eds.), American Mathematical Society
    • W-T. Wu, "On the decision problem and the mechanization of theorem-proving in elementary geometry," in Automated Theorem Proving: After 25 Years, W. Bledsoe and D. Loveland (Eds.), American Mathematical Society, 1983, pp. 213-234.
    • (1983) Automated Theorem Proving: After 25 Years , pp. 213-234
    • Wu, W.-T.1
  • 118
    • 0344495218 scopus 로고
    • Abzählende Methoden
    • Leipzig, dritter Band, zweiter Teil, erste Hälfte
    • H.G. Zeuthen, "Abzählende Methoden," in Enzyklopädie der Mathematischen Wissenschaften, Leipzig, 1903-1915, pp. 43-87. dritter Band, zweiter Teil, erste Hälfte.
    • (1903) Enzyklopädie der Mathematischen Wissenschaften , pp. 43-87
    • Zeuthen, H.G.1


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