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Square-root Parametrization of Plane Curves

  • Shreeram S. Abhyankar

Abstract

By calculating the genus of an irreducible algebraic plane curve of degree n in terms of its singularities, we see that, counted properly, the curve can have at most \(\frac{{(n - 1)(n - 2)}} {2}\) double points, and it can be rationally parametrized iff this maximum is reached. If the maximum falls short by one or two, then the curve can still be parametrized by square-roots of rational functions. Such a square-root parametrization is used for factoring certain bivariate polynomials over a finite field.

Keywords

Galois Group Double Point Plane Curf Hyperelliptic Curve Splitting Field 
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. [1]
    S. S. Abhyankar, Coverings of algebraic curves, American Journal of Mathematics, vol 79, 1957, pages 825–856.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    S. S. Abhyankar, Historical ramblings in algebraic geometry and related algebra, American Mathematical Monthly, vol 83, 1976, pages 409–448.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    S. S. Abhyankar, Algebraic Geometry for Scientists and Engineers, American Mathematical Society, Providence, 1990.Google Scholar
  4. [4]
    S. S. Abhyankar, Galois theory on the line in nonzero characteristic, Bulletin of the American Mathematical Society, (to appear).Google Scholar
  5. [5]
    S. S. Abhyankar and C. Bajaj, Automatic parametrization of curves and surfaces III: Algebraic plane curves, Computer Aided Geometric Design, vol 5, 1988, pages 309–321.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    S. S. Abhyankar and C. Bajaj, Computations with algebraic curves, Springer Lecture Notes In Computer Science, vol 358, 1989, pages 274–284.MathSciNetGoogle Scholar
  7. [7]
    J-P. Serre, A letter as an appendix to the square-root paper of Abhyankar, These Proceedings.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1994

Authors and Affiliations

  • Shreeram S. Abhyankar

There are no affiliations available

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