Rational parametrization of surfaces

Journal of Symbolic Computation

Volumn 26, Issue 1, 1998, Pages 1-29

  Schicho, Josef ^a  

^a JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032119594     PISSN: 07477171     EISSN: None     Source Type: Journal    
DOI: 10.1006/jsco.1997.0199     Document Type: Article

References (60)

1. Abhyankar, S. S. (1969). Resolution of singularities of algebraic surfaces. In Algebraic Geometry, pp. 1-11. Oxford University Press.
2. Abhyankar, S. S., Bajaj, B. (1987). Automatic parametrization of curves and surfaces III. Comp. Aided Geom. Des., 5-4, 309-323.
3. Alonso, M. E., Luengo, I., Raimondo, M. (1988). An algorithm on quasi-ordinary polynomials. Appl. Algebra Eng. Commun. Comput., 6, 59-73.
4. Becker, T., Weispfenning, V. (1993). Gröbner Bases - A Computational Approach to Commutative Algebra. Graduate Texts in Mathematics. Springer Verlag.
5. Bierstone, E., Milman, P. (1991). A simple constructive proof of canonical resolution of singularities. In Lopez, A. C., Macarro, L. N., eds, Effective Methods in Algebraic Geometry, pp. 11-30. Birkhauser.
6. Blass, P., Lipman, J. (1979). Remarks on adjoints and arithmetic genera of algebraic varieties. Am. J. Math., 101, 331-336.
7. Böhm, W., Farin, G., Kahmann, J. (1984). A survey of curve and surface methods in cagd. Comp. Aided Geom. Design, 1, 1-60.
8. Brown, W. S., Traub, J. F. (1971). On Euclid's algorithm and the theory of subresultants. Journal ACM, 18, 505-514.
9. Buchberger, B. (1965). An Algorithm for Finding a Basis for the Residue Class Ring of a Zero-Dimensional Polynomial Ideal. PhD thesis, Universitat Innsbruck, Institut fur Mathematik. German.
10. Buchberger, B. (1985). Groebner bases: an algorithmic method in polynomial ideal theory. In Bose, N. K., ed., Recent Trends in Multidimensional Systems Theory, chapter 6. D. Riedel Publ. Comp.
11. Buchberger, B., Collins, G. E., Loos, R., eds, (1982). Computer Algebra: Symbolic and Algebraic Computation. Springer, Vienna-New York, 2nd edn.
12. Canny, J., Manocha, D. (1992). Implicit representation of rational parametric surfaces. J. Symb. Comput., 13(5).
13. Castelnuovo, G. (1939). Sulle superficie di genere zero. In Memoria Scelte, pp. 137-187. Zanichelli.
14. Chionh, E.-W., Goldman, R. (1992). Degree, multiplicity and inversion formulae for rational surfaces using u-resultants. Comp. Aided Geom. Des., 9, 93-108.
15. Clebsch, A. (1868). Über die Abbildungen algebraischer Flächen, insbesondere der vierten und fünften Ordnung. Math. Ann., 1, 253-316.
16. Collins, G. E. (1967). Subresultants and reduced polynomial remainder sequences. Journal ACM, 14, 128-142.
17. Conforto, F. (1939). La Superfici Razionali. Zanichelli.
18. Eisenbud, D. (1993). Open problems in computational algebraic geometry. In Cortona 1991, volume 34 of Proc. Symp. Math., pp. 49-70.
19. Enriques, F. (1895). Sulle irrazionalita da cui puo farsi dipendere la risoluzione d'un equazione f(xyz)=0 con funzioni razionali di due parametri. Math. Ann., 1-23.
20. Enriques, F. (1949). La Superfici Algebriche. Zanichelli.
21. Farin, G. F. (1988). Curves and Surfaces for Computer Aided Geometric Design: A practical guide. Academic Press.
22. Gao, X.-S., Chou, S.-C. (1992). Implicitization of rational parametric equations. J. Symb. Comput., 14, pp. 459-470.
23. Grauert, H., Riemenschneider, O. (1970). Verschwindungssätze für analytische Kohomologiegruppen. Invent. Math., 11, 263-292.
24. Greenberg, M. J. (1969). Lectures on Forms in Many Variables. Benjamnin.
25. Griffiths, P., Harris, J. (1978). Principles of Algebraic Geometry. John Wiley.
26. Hartshorne, R. (1977). Algebraic Geometry. Springer-Verlag.
27. Hillgarter, E. (1996). Rational points on conics. Master's thesis, RISC Linz.
28. Hironaka, H. (1984). Desingularization of excellent surfaces. In Gossart, V., Giraud, J., Orbanz, U., eds, Resolution of Surface Singularities, pp. 99-132. Springer.
29. Jung, H. (1908). Darstellung der Funktionen eines algebraischen Körpers zweier unabhängiger Veränderlicher. J. Reine Angew. Math., 133, 289-314.
30. Kalkbrener, M. (1990). Implicitization of Rational Curves and Surfaces. In Sakata, ed., Lect. Notes in Comp. Sci. 508, AAECC-8, Tokyo, Japan.
31. Kalkbrener, M. (1995). An algorithm for solving systems of algebraic equations in three variables. In Pfalzgraf, J., Wang, D., eds, Automated Practical Reasoning, pp. 81-90. Springer.
32. Kurke, H. (1982). Vorlesungen über Algebraische Flächen. Teubner.
33. Lipman, J. (1978). Desingularization of two-dimensional schemes. Ann. Math., 107, 151-207.
34. Mñuk, M. (1996). Algebraic parametrization of rational curves. PhD thesis, RISC Linz.
35. Mñuk, M., Sendra, J. R., Winkler, F. (1996). On the complexity of parametrizing curves. Beitr. Algebra und Geometrie, 37/2.
36. Manin, Y. (1974). Cubic Forms. North-Holland.
37. Mishra, B. (1993). Algorithmic Algebra. Springer, New York.
38. Nöether, M. (1871). Zur Theorie eineindeutigen Entsprechens höherer Raumgebilde. Math. Ann., 4.
39. Pfalzgraf, J., Wang, D. (1995). Automated Practical Reasoning. Springer.
40. Pottmann, H. (1996). Applications of laguerre geometry in cagd. Technical Report 30,31, Techn. Univ. Wien.
41. Qiulin, D., Davies, B. J. (1987). Surface Engineering Geometry for Computer-aided Design and Manufacture. John Wiley.
42. Reider, I. (1988). Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. Math., 127, 309-316.
43. Schicho, J. (1992). On the choice of pencils in the parametrization of curves. J. Symb. Comput., 14, 557-576.
44. Schicho, J. (1995). Rational Parametrization of Surfaces. PhD thesis, RISC Linz.
45. Sederberg, T. W., Snively, J. P. (1987). Parametrization of cubic algebraic surfaces. In The Mathematics of Surfaces, pp. 299-341. Clarendon Press.
46. Sendra, J., Winkler, F. (1991). Symbolic parametrization of curves. J. Symb. Comput., 12, 607-632.
47. Sendra, J., Winkler, F. (1997). Parametrization of algebraic curves over optimal field extensions. J. Symb. Comput., 23, 191-208.
48. Shafarevich, I. R., ed. (1965). Algebraic Surfaces. Proc. Steklov Inst. Math. transl. by AMS 1967.
49. Shafarevich, I. R. (1974). Algebraic Geometry. Springer.
50. Teitelbaum, J. (1990). The computational complexity of the resolution of plane curve singularities. Math. Comp., 54.
51. Tran, Q.-N., Winkler, F. (1997). CASA Reference Manual. http://www.risc.uni-linz.ac.at/software/casa.
52. van Hoeij, M. (1994). Computing parametrizations of rational algebraic curves. In ISSAC-94, pages 187-190. ACM Press.
53. van Hoeij, M. (1997). Rational parametrizations of rational algebraic curves. J. Symb. Comput., to appear.
54. Villamayor, O. (1989). Constructiveness of Hironaka's resolution. Ann. Scient. Ecole Norm. Sup. 4, 22, 1-32.
55. Volcheck, E. (1995). How to compute the adjoint divisor of a curve over a finite field. preprint.
56. Walker, R. J. (1935). Reduction of the singularities of an algebraic surface. Ann. Math., 36, 336-365.
57. Walker, R. J. (1978). Algebraic Curves. Springer.
58. Wu, W. T. (1986). Basic principles of mechianical theorem proving in elementary geometries. J. Autom. Reasoning, 2, 221-252.
59. Zariski, O. (1939). The reduction of singularities of an algebraic surface. Ann. Math., 40, 639-689.
60. Zariski, O. (1971). Algebraic Surfaces. Springer, 2nd edn.