On the isomorphisms of smooth algebraic curves

ABSTRACT

In this paper, I have considered some problems of algebraic curves in some constructive way, especially, I give an algorithm for determining whether two given smooth plane curves are isomorphic and finding all isomorphic maps. I also have given a survey of some miscellaneous results related to the classification of curves. In the appendix, I give some other results which implies a more efficient algorithm for deciding whether two plane curves are isomorphic and finding all isomorphic maps. It is clear our method in this paper can be generalized to smooth projective complete intersection varieties.

REFERENCES

• 1: H. Du, On the isomorphisms of algebraic curves (I), MMRC Research Reports, No. 9, 1993, Institute of Systems Science, Academia Sinica.
• 2: H. Du, On the isomorphisms of hypersurfaces, MMRC Research Reports, No. 10, 1993, Institute of Systems Science, Academia Sinica.
• 3: H.Du, On the isomorphisms of complete intersection varieties, MMRC Research Reports, No. 10, 1993, Institute of Systems Science, Academia Sinica.
• 4: H. Du, J. Wu, On the computation of zero-dimensional ideal, MMRC Research Reports, No. 10, 1993, institute of Systems Science, Academia Sinica.
• 5: H. Du, On the constructive problems of algebraic curves. Preprint.
• 6: D. Eisenbud, From the view point of commutative algebra towards algebraic geometry. (a book in preparation)
• 7: P. Griffiths, J. Harris, Principle of Algebraic Geometry. John Wiley & Sons, Inc. 1978
• 8: R.Hartshorne,Algebraic geometry, Springer verlag, 1982
• 9: D. Mumford, Geometric invariant theory, Springer verlag, 1965
• 10: J. H. Silverman, Arithmetic of elliptic curves, Springer verlag, 1986
• 11: Wen-tsün Wu, Mechanical theorem proving in geometries, Springer-Verlag New York, Inc., New York, NY, 1994

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Algebraic algorithms

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Geometrical problems and computations

General Terms:
Algorithms, Theory, Verification