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CASA: Computer Algebra Software for Computing with Algebraic Sets

  • Bernhard Wall
Part of the Texts and Monographs in Symbolic Computation book series (TEXTSMONOGR)

Abstract

In this report we want to illustrate with two examples how the program package “Computer Algebra Software for Constructive Algebraic Geometry” (CASA) (see Gebauer et al. 1991) can be used in order to reason about geometric objects defined by algebraic equations.

Keywords

Algebraic Curve Algebraic Curf Power Series Expansion Birational Mapping Newton Polygon 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Abhyankar, S. S. (1975): Approximate roots of polynomials and special cases of the epimorphism theorem. Lecture notes by Chris Christensen, Division of Mathematical Sciences, Purdue University, West Lafayette, IN.Google Scholar
  2. Abhyankar, S. S., Bajaj, C. (1988a): Automatic parametrization of rational curves and surfaces IV: algebraic space curves. Res. Rep., Computer Science Department, Purdue University, West Lafayette, IN.Google Scholar
  3. Abhyankar, S. S., Bajaj, C. (1988b): Automatic parametrization of rational curves and surfaces III: algebraic plane curves. Comput. Aided Geom. Des. 5: 309–321.MathSciNetMATHCrossRefGoogle Scholar
  4. Abhyankar, S. S., Bajaj, C. (1989): Computations with algebraic curves. In: Gianni, P. (ed.): Symbolic and algebraic computation. Springer, Berlin Heidelberg New York Tokyo, pp. 274–284 (Lecture notes in computer science, vol. 358).CrossRefGoogle Scholar
  5. Bajaj, C., Royappa, A. (1989): GANITH: an algebraic geometry package. Res. Rep. csd-tr-914, Computer Science Department, Purdue University, West Lafayette, IN.Google Scholar
  6. Bajaj, C., Royappa, A. (1990): The GANITH algebraic geometry toolkit. In: Miola, A. (ed.): Design and implementation of symbolic computation systems. Springer, Berlin Heidelberg New York Tokyo, pp. 268–269 (Lecture notes in computer science, vol. 429).CrossRefGoogle Scholar
  7. Bajaj, C., Hoffmann, C. M., Lynch, R. E., Hopcroft, J. E. H. (1988): Tracing surface intersections. Comput. Aided Geom. Des. 5: 285–307.MathSciNetMATHCrossRefGoogle Scholar
  8. Bennett, D. (1990): Interactive display and manipulation of curves and surfaces of mathematical functions. ACM SIGSAM Bull. 24(3): 33–34.Google Scholar
  9. Brieskorn, E., Knörrer, H. (1986): Plane algebraic curves. Birkhäuser, Basel.MATHGoogle Scholar
  10. Char, B. W., Geddes, K. O., Gonnet, G. H., Leong, B. L., Monagan, M. B., Watt, S. M. (1991a): Maple V language reference manual. Springer, Berlin Heidelberg New York Tokyo.MATHCrossRefGoogle Scholar
  11. Char, B. W., Geddes, K. O., Gonnet, G. H., Leong, B. L., Monagan, M. B., Watt, S. M. (1991b): Maple V library reference manual. Springer, Berlin Heidelberg New York Tokyo.MATHCrossRefGoogle Scholar
  12. Char, B. W., Geddes, K. O., Gonnet, G. H., Leong, B. L., Monagan, M. B., Watt, S. M. (1992): First leaves: a tutorial introduction to Maple V. Springer, Berlin Heidelberg New York Tokyo.MATHCrossRefGoogle Scholar
  13. Duval, D. (1987): Diverses questions relatives au calcul formel avec des nombres algébriques. Ph.D. thesis, Institut Fourier, Grenoble, France.Google Scholar
  14. Duval, D. (1989): Rational Puiseux expansion. Compos. Math. 70: 119–154.MathSciNetMATHGoogle Scholar
  15. Eichler, M. (1966): Introduction to the theory of algebraic numbers and functions. Academic Press, London.MATHGoogle Scholar
  16. Gebauer, R. (1990): Plotting algebraic curves. Tech. Rep. RISC Linz 90-69.0.Google Scholar
  17. Gebauer, R., Kalkbrener, M., Wall, B., Winkler, F. (1991): CASA: a computer algebra package for constructive algebraic geometry. In: Watt, S. M. (ed.): ISSAC’ 91, Bonn, Germany, July 1991, pp. 403–410.Google Scholar
  18. Henry, J. P. G., Merle, M. (1987a): Complexity of computation of embedded resolution of algebraic curves. In: Davenport, J. H. (ed.): EUROCAL’ 87. Springer, Berlin Heidelberg New York Tokyo, pp. 381–390 (Lecture notes in computer science, vol. 378).Google Scholar
  19. Henry, J. P. G., Merle, M. (1987b): Puiseux pairs, resolution of curves and lazy evaluation. Res. Rep., Centre de Mathématiques, École Polytechniques, Palaiseau.Google Scholar
  20. Hoffmann, C. M. (1988): Algebraic curves. In: Rice, J. (ed.): Mathematical aspects of scientific software. Springer, Berlin Heidelberg New York Tokyo, pp. 101–122 (The IMA volumes in mathematics and its applications, vol. 14).CrossRefGoogle Scholar
  21. Hoffmann, C. M. (1989): Geometric and solid modelling — an introduction. Morgan Kauffmann, San Mateo, CA.Google Scholar
  22. Kalkbrener, M. (1990a): Birational projections of irreducible varieties. Tech. Rep. RISC Linz 90-59.0.Google Scholar
  23. Kalkbrener, M. (1990b): Implicitization by using Gröbner bases. Tech. Rep. RISC Linz 90-27.0.Google Scholar
  24. Kalkbrener, M. (1991a): Implicitization of rational curves and surfaces. In: Sakata, S. (ed.): Applied algebra, algebraic algorithms, error-correcting codes. Springer, Berlin Heidelberg New York Tokyo, pp. 249–259 (Lecture notes in computer science, vol. 508).CrossRefGoogle Scholar
  25. Kalkbrener, M. (1991b): Three contributions to elimination theory. Ph.D. thesis, Research Institute for Symbolic Computation, University of Linz, Linz, Austria.Google Scholar
  26. Kredel, H., Weispfenning, V. (1988): Computing dimension and independent sets for polynomial ideals. J. Symb. Comput. 6: 231–248.MathSciNetMATHCrossRefGoogle Scholar
  27. Kung, H. T., Traub, J. F. (1978): All algebraic functions can be computed fast. J. ACM 25: 245–260.MathSciNetMATHCrossRefGoogle Scholar
  28. Milnor, J. (1968): Singular points of complex hypersurfaces. Princeton University Press, Princeton.MATHGoogle Scholar
  29. Mora, F. (1982): An algorithm to compute the equation of tangent cones. In: Calmet, J. (ed.): Computer algebra. Springer, Berlin Heidelberg New York Tokyo, pp. 273–280 (Lecture notes in computer science, vol. 144).Google Scholar
  30. Newton, I. (1969): Methods of series and fluxions. In: Whiteside, D. T. (ed.): The mathematical papers of Isaac Newton. Cambridge University Press, Cambridge.Google Scholar
  31. Puiseux, M. V. (1850): Recherches sur les fonctions algebriques. J. Math. Pures Appl. 15: 365–480.Google Scholar
  32. Ritt, J. F. (1950): Differential algebra. American Mathematical Society, New York.MATHGoogle Scholar
  33. Rybowicz, M. (1990): Sur le calcul des places et des anneaux d’entiers d’un corps de fonctions algebriques. Ph.D. thesis, Université de Limoges, Limoges, France.Google Scholar
  34. Sendra, J. R., Winkler, F. (1991): Symbolic parametrization of curves. J. Symb. Comput. 12: 607–632.MathSciNetMATHCrossRefGoogle Scholar
  35. Stillman, M., Stillman, M., Bayer, D. (1989): Macaulay user manual. Available via FTP on site 128.103.28.10 (math.harvard.edu).Google Scholar
  36. Stobbe, R. (1992): Darstellung algebraischer Kurven mittels Computergrafik. Master’s thesis, Fachbereich Mathematik der Universität Kaiserslautern, Kaiserslautern, Germany.Google Scholar
  37. van der Waerden, B. L. (1973): Einführung in die algebraische Geometrie, 2nd edn. Springer, Berlin Heidelberg New York.MATHGoogle Scholar
  38. Walker, R. J. (1978): Algebraic curves, 2nd edn. Springer, Berlin Heidelberg New York.MATHCrossRefGoogle Scholar
  39. Wall, B. (1991a): CASA: computer algebra software for constructive algebraic geometry in Maple — a primer. Tech. Rep. RISC Linz 91-34.1.Google Scholar
  40. Wall, B. (1991b): Puiseux expansion: an annotated bibliography. Tech. Rep. RISC Linz 91-46.0.Google Scholar
  41. Wang, D. (1989): A method for determining the finite basis of an ideal from its characteristic set with application to irreducible decomposition of algebraic varieties. Tech. Rep. RISC Linz 89-50.0.Google Scholar
  42. Wang, D. (1992): Irreducible decomposition of algebraic varieties via characteristic sets and Gröbner bases. Comput. Aided Geom. Des. 9: 471–484.MATHCrossRefGoogle Scholar
  43. Wu, W. (1984): Basic principles of mechanical theorem proving in elementary geometries. J. Syst. Sci. Math. Sci. 4: 207–235.Google Scholar

Copyright information

© Springer-Verlag Wien 1995

Authors and Affiliations

  • Bernhard Wall

There are no affiliations available

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