Computer algebra and algebraic. Geometry - Achievements and perspectives

Journal of Symbolic Computation

Volumn 30, Issue 3, 2000, Pages 253-289

  Greuel, Gert Martin ^a  

^a France   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034258731     PISSN: 07477171     EISSN: None     Source Type: Journal    
DOI: 10.1006/jsco.2000.0362     Document Type: Article

References (104)

1. Adams, W. W., Loustaunou, P. (1994). An introduction to Gröbner bases. In AMS Graduate Studies in Mathematics 3, Providence, R.I.
2. Apel, J. (1998). Computational Ideal Theory in Finitely Generated Extension Rings. Preprint, Leipzig.
3. Apel, J., Klaus, U. (1991). FELIX - an assistant for algebrists. In Proceedings of ISSAC'91, pp. 382-389.
4. Arnold, V. I., Gusein-Zade, S. M., Varchenko, A. N. (1995). Singularities of Differential Maps, volume I, Birkhäuser.
5. Bayer, D. (1982). The Division Algorithm and the Hilbert-scheme, Thesis. Harvard University.
6. Bayer, D., Stillman, M. (1982-1990). Macaulay: A System for Computation in Algebraic Geometry and Commutative Algebra. Available via ftp from zariski.harvard.edu.
7. Bayer, T. (2000). Computation of Moduli Spaces for Semiquasihomogeneous Singularities and an Implementation in SINGULAR. Kaiserslautern, Diplomarbeit.
8. Becker, E., Wörmann, T. (1996). Radical computations of zero-dimensional ideals and real root counting. Mathematics and Computers in Simulation, 42, 561-569.
9. Becker, T., Weispfennig, V. (1993). Gröbner bases, a computational approach to commutative algebra, Graduate Texts in Mathematics 141, Springer.
10. Brieskorn, E. (1970). Die Monodromie der Isolierten Singularitäten von Hyperflächen. Manuscripta Math., 2, 103-161.
11. Buchberger, B. (1965). Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem Nulldimensionalen Polynomideal. Ph.D. Thesis. University of Innsbruck, Austria.
12. Buchberger, B. (1970). Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen gleichungssystems. Äqu. Math., 4, 374-383.
13. Buchberger, B. (1985). Gröbner bases: An algorithmic method in polynomial ideal theory. In Bose, N. B. ed., Recent Trends in Multidimensional System Theory. Reidel.
14. (1998). In Buchberger, B., Winkler, F. eds, Gröbner Bases and Applications, LNS 251, pp. 109-143, CUP.
15. Canny, J. F., Emiris, I. Z. (1997). A Subdivision-based Algorithm for the Sparse Resultant. Preprint, Berkeley.
16. In Cohen, A. M., Cuypers, H., Sterk, H. Eds, (1999). Some Tapas of Computer Algebra, Springer-Verlag.
17. Cox, D., Little, J., O'Shea, D. (1992). Ideals, Varieties and Algorithms. Springer-Verlag.
18. Cox, D., Little, J., O'Shea, D. (1998). Using Algebraic Geometry. Springer-Verlag.
19. Capani, A., Niesi, G., Robbiano, L., CoCoA, (1995). Available via http:// lancelot.dima.unige.it.
20. Davenport, J. H., Siret, Y., Tournier, E. (1988). Computer algebra. In Systems and Algorithms for Algebraic Computations, 2nd Edn, Academic Press.
21. Decker, W., De Jong, T. (1998). Gröbner bases and invariant theory. In Buchberger, B., Winkler, F. eds, Gröbner Bases and Applications, LNS 251, pp. 61-89, CUP.
22. Decker, W., Greuel, G.-M., Pfister, G. (1998). Primary decomposition: Algorithms and comparisons. In Greuel, G.-M., Matzat, H. B., Hiss, G. eds, Algorithmic Algebra and Number Theory, pp. 187-220. Heidelberg, Springer-Verlag.
23. 4. J. Algebraic Geometry, 2, 185-237.
24. Decker, W., Greuel, G.-M., De Jong, T., Pfister, G. (1998). The normalisation: A new algorithm, implementation and comparisons. In Proc. Computational Methods for Representations of Groups and Algebras, pp. 177-185. Birkhäuser.
25. Deligne, P. (1970). Equations Differentielles a Points Singuliers Reguliers, LNM 163. Springer.
26. Derksen, H. (1997). Constructive Invariant Theory and the Linearization Problem, Thesis. Basel.
27. Eisenbud, D. (1993). Open problems in computational algebraic geometry and commutative algebra. In Eisenbud, D., Robbiano, L. eds, Computational Algebraic Geometry and Commutative Algebra, Cortona 1991, pp. 49-71. Cambridge, England, Cambridge University Press.
28. Eisenbud, D. (1995). Commutative Algebra with a View Toward Algebraic Geometry. Springer Verlag.
29. Eisenbud, D., Sturmfels, B. (1996). Binomial ideals. Duke Math. J., 84, 1-45.
30. Eisenbud, D., Diaconis, P., Sturmfels, B. (1996). Lattice walks and primary decomposition. Rotafest Proceedings, to appear.
31. Eisenbud, D., Huneke, C., Vasconcelos, W. (1992). Direct methods for primary decomposition. Invent. Math 110, 207-235.
32. Eisenbud, D., Peeva, I., Sturmfels, B. Noncommutative Gröbner bases for commutative ideals. Proc. Am. Math. Soc., to appear.
33. Fröberg, R. (1997). An Introduction to Gröbner Bases. John Wiley & Sons.
34. Frühbis-Krüger, A. (2000). Construction of Moduli Spaces for Space Curve Singularities of Multiplicity 3. Preprint, Kaiserslautern.
35. Gatermann, K. (1999). Computer Algebra Methods for Equivariant Dynamical Systems. Preprint SC, Berlin, 99-26.
36. Von zur Gathen, J., Gerhard, J. (1999). Modern Computer Algebra. Cambridge University Press.
37. Gelfand, I., Kapranov, M., Zelevinski, A. (1994). Discriminants, Resultants and Multidimensional Determinants. Birkhäuser Verlag.
38. Gerard, R., Levelt, A. H. M. (1973). Invariants mesurant l'irregularite en un point singulier des systemes de'equations differentielles lineares. Ann. Inst. Fourier Grenoble, 23, 157-195.
39. Gianni, P., Trager, B. (1997). Integral closure of noetherian rings. In Proceedings ISSAC 97.
40. Gianni, P., Trager, B., Zacharias, G. (1988). Gröbner bases and primary decomposition of polynomial ideals. J. Symb. Comp., 6, 149-167.
41. Grabmeier, J., Kaltofen, E., Weispfennig, V. (2000). Handbook of Computer Algebra, Foundations, Applications, Systems, Springer-Verlag, to appear.
42. Gräbe, H.-G. (1995). Algorithms in local algebra. J. Symb. Comp., 19, 545-557.
43. Grassmann, H., Greuel, G.-M., Martin, B., Neumann, W., Pfister, G., Pohl, W., Schönemann, H., Siebert, T. (1994). Standard bases, syzygies and their implementation in SINGULAR. In Beiträge zur Angewandten Analysis und Informatik, pp. 69-96. Aachen, Shaker.
44. Grauert, H., Remmert, R. (1971). Analytische Stellenalgebren. Springer-Verlag.
45. Grayson, D., Stillmann, M. (1996). Macaulay2, a Computer Software System Designed to Support Research in Commutative Algebra and Algebraic Geometry. Available from http:// math.uiuc.edu/Macaulay2.
46. Greuel, G.-M. (1975). Der Gauss-Manin zusammenhang isolierter singularitäten von vollständigen durchschnitten. Math. Ann., 214, 235-266.
47. Greuel, G.-M. (1992). Deformation und Klassifikation von Singularitäten und Moduln. Jahresber. Deutsch. Math.-Verein. Jubiläumstagung 1990, 177-238.
48. Greuel, G.-M. (1986). Constant Milnor number implies constant multiplicity for quasihomogeneous singularities. Manuscripta Math., 56, 159-166.
49. Greuel, G.-M., Kröning, H. (1990). Simple singularities in positive characteristic. Math. Z., 203, 339-354.
50. Greuel, G.-M., Pfister, G. (1996). Advances and improvements in the theory of standard bases and syzygies. Arch. Math., 66, 163-1796.
51. Greuel, G.-M., Pfister, G. (1998). Gröbner bases and algebraic geometry. In Buchberger, B., Winkler, F. eds, Gröbner Bases and Applications, LNS 251, pp. 109-143, CUP.
52. Greuel, G.-M., Lossen, C., Shustin, E. (1998). Plane curves of minimal degree with prescribed singularities. Invent. Math., 133, 539-580.
53. Greuel, G.-M., Martin, B., Pfister, G. (1985). Numerische Charakterisierung quasihomogener Gorenstein-Kurvensingularitäten. Math. Nachr., 124, 123-131.
54. Greuel, G.-M., Pfister, G., Schönemann, H. (1990-1998). Singular, A system for polynomial computations, version 1.2 user manual. In: Reports on Computer Algebra, Number 21. Centre for Computer Algebra, University of Kaiserslautern, June 1998. Available via http://www.mathematik.uni-kl.de/zca/Singular.
55. Hillebrand, D. (1999). Triangulierung Nulldimensionaler Ideale - Implementierung und Vergleich Zweier Algorithmen, Dortmund, Diplomarbeit.
56. Hironaka, H. (1964). Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. Math., 79, 109-326.
57. ISSAC (1988-1998). In International Symposium on Symbolic and Algebraic Computation, Proceedings.
58. De Jong, T. (1998). An algorithm for computing the integral closure. J. Symb. Comp., 26, 273-277.
59. Kalkbrenner, M. (1994). Prime decomposition of radicals in polynomial rings. J. Symb. Comp., 18, 365-372.
60. Kalkbrenner, M. (1998). Algorithmic properties of polynomial rings. J. Symb. Comp., 26, 525-581.
61. Kandri-Rody, A., Weispfenning, V. (1990). Non-cummutative Gröbner bases in algebras of solvable type. J. Symb. Comput., 9, 1-26.
62. Kapur, D., Madlener, K. (1989). A completion procedure for computing a canonical basis for a k- Subalgebra. In Kaltofen, E., Watt, S. M. eds, Computer and Mathematics, pp. 1-11. Springer.
63. Kemper, G. (1996). Calculating invariant rings of finite groups over arbitrary fields. J. Symb. Comp., 21, 351-366.
64. Kontsevich, M., Manin, Y. (1994). Gromov-Witten classes, quantum cohomology, and enumerative geometry. Comm. Math. Phys., 164, 525-562.
65. Krick, T., Logar, A. (1991). An algorithm for the computation of the radical of an ideal in the ring of polynomials. In AAECC9, LNCS 539, pp. 195-205. Springer.
66. Krüger, K. (1997). Klassifikation Von Hyperflächensingularitäten. Kaiserslautern, Diplomarbeit.
67. Lamm, M. (1999). Hamburger-noether-entwicklung Von Kurvensingularitäten. Kaiserslautern, Diplomarbeit.
68. Lossen, C. (1999). New asymptotics for the existence of plane curves with prescribed singularities. Comm. Alg., 27, 3263-3282.
69. Laudal, A. (1979). Formal Moduli of Algebraic Structures. SLN 754, Springer-Verlag.
70. Martin, B. (1996). Computing Massey Products Using SINGULAR. Preprint M-02, Cottbus.
71. Martin, B. (1998). Computing versal deformations with SINGULAR. In Matzat, B. H., Greuel, G.-M., Hiss, G. eds, Algorithmic Algebra and Number Theory, pp. 283-294. Heidelberg, Springer-Verlag.
72. Martin, B., Pfister, G. (1989). The kernel of the Kodaira-Spencer map of the versal μ-constant deformation of an irreducible plane curve singularity with ℂ*-action. J. Symb. Comp., 7, 527-539.
73. Matzat, B. H., Greuel, G.-M., Hiss, G., Eds, (1998). Algorithmic Algebra and Number Theory, Heidelberg, Springer-Verlag.
74. Mishra, B. (1993). Algorithmic Algebra, Texts and Monographs in Computer Science. Springer.
75. Möller, H. M. (1998). Gröbner bases and numerical analysis. In Buchberger, B., Winkler, F. eds, Gröbner Bases and Applications, LNS 251, pp. 159-178, CUP.
76. Mora, T. (1982). An algorithm to compute the equations of tangent cones. In Proc. EUROCAM 82, Lecture Notes in Computer Science.
77. Mora, T. (1989). In Gröbner Bases in Non-commutative Algebras, LNCS 358, pp. 150-161. Springer-Verlag.
78. Mora, T., Pfister, G., Traverso, C. (1992). An introduction to the tangent cone algorithm. Advances in computing research. In Robotics and Non-linear Geometry, 6, 199-270.
79. O'Shea, D. (1987). Topologically trivial deformations of isolated quasihomogeneous hypersurface singularities are equimultiple. In Proc. AMS 101, pp. 260-262.
80. Pfister, G. (1991). The tangent cone algorithm and some applications to algebraic geometry. In Proceedings of the MEGA Conference 1990. Birkhäuser.
81. Pfister, G., Schönemann, H. (1989). Singularities with exact Poincarè complex but not quasihomogeneous. Revista Mathematica Madrid (2) 2y3, 161-171.
82. Ritt, J. F. (1950). Differential algebra. In Colloquium Publications, volume 33, New York, AMS.
83. Robbiano, L., Sweedler, M. (1990). In Bruns, W., Simis, A. eds, Subalgebra Bases, Springer LNM, 1430, pp. 61-87.
84. Saito, K. Quasihomogene isolierte singularitäten von hyperflächen. Invent. Math., 14, 123-142.
85. Saito, M., Sturmfels, B., Takayama, N. (1999). Gröbner Deformations of Hypergeometric Differential Equations. Springer-Verlag.
86. Schreyer, F.-O. (1980). Die Berechnung von Syzygien mit dem Verallgemeinerten Weierstrass'schen Divisionssatz. Hamburg, Diplomarbeit.
87. Schreyer, F.-O. (1991). A standard basis approach to syzygies of canonical curves. J. Reine Angew. Math., 421, 83-123.
88. Schreyer, F.-O. (1986). Syzygies of canonical curves and special linear series. Math. Ann., 275.
89. Schulze, M. (1999). Computation of the Monodromy of an Isolated Hypersurface Singularity, Diploma Thesis. Kaiserslautern.
90. Seidenberg, A. (1975). Construction of the integral closure of a finite integral domain II. Proc. Amer. Math. Soc., 52, 368-372.
91. Shimoyama, T., Yokoyama, K. (1996). Localization and primary decomposition of polynomial ideals. J. Symb. Comp., 22, 247-277.
92. Stetter, H. (1996). Matrix eigenproblems are at the heart of polynomial systems solving. Sigsam Bull., 30, 22-25.
93. Stetter, H. (1997). Stablization of polynomial systems solving with Groebner bases. In ISSAC 97 Proceedings, pp. 117-124.
94. Stolzenberg, G. (1968). Constructive normalization of an algebraic variety. Bull. Amer. Math. Soc., 74, 595-599.
95. Sturmfels, B. (1993). Algorithms in Invariant Theory. Springer-Verlag.
96. Ufnarovski, V. (1998). Introduction to noncommutative Gröbner bases theory. In Buchberger, B., Winkler, F. eds, Gröbner Bases and Applications, LNS 251, pp. 259-280, CUP.
97. Vasconcelos, V. W. (1991). Computing the integral closure of an affine domain. Proc. AMS, 113 (3), 633-638.
98. Vasconcelos, V. W. (1998). Computational Methods in Commutative Algebra and Algebraic Geometry. Springer-Verlag.
99. Verschelde, J. (1999). Polynomial Homotopies for Dense, Sparse and Determinantal Systems. Preprint Math.NA/9907060.
100. Wall, C. T. C. (1983). Classification of unimodal isolated singularities of complete intersections. In: Singularities, Arcata 1981 (Ed. Orlik, P.). Proc. Sympos. Pure Math., 40 (2), 625-640.
101. Wang, D. (1992). Irreducible decomposition of algebraic varieties via characteristic sets and Gröbner bases. Computer Aided Geometric Design, 9, 471-484.
102. Weispfenning, V. (1992). Comprehensive Gröbner bases. J. Symb. Comput., 14, 1-29.
103. Wenk, M. (1999). Resultantenmethoden zur Lösung Algebraischer Gleichungssysteme Implementiert in SINGULAR. Kaiserslautern, Diplomarbeit.
104. Zariski, O. (1971). Some open questions in the theory of singularities. Bull. AMS, 481-491.