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Polynomial parametrization of nonsingular complete intersection curves

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Abstract

In this paper we give a new method to compute a polynomial parametrization, in case it exists, of an affine nonsingular complete intersection curve. Our method is based on the study of vector fields on nonsingular algebraic curves. In contrast to the existing methods, our algorithm does not use any projection, and no sample point on the curve is needed. It is also important to stress the fact that the algorithm produces a parametrization with coefficients in the ground field.

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References

  1. Abhyankar S. (1988). What is the difference between a parabola and a hyperbola. Math. Intell. 10(4): 36–43

    Article  MATH  MathSciNet  Google Scholar 

  2. Abhyankar, S.: Algebraic geometry for scientists and engineers, vol 35. Math. Surv. Monogr. AMS (1990)

  3. Abhyankar S. and Bajaj C. (1989). Automatic parameterization of rational curves and surfaces, iv: algebraic space curves. ACM Trans. Graph. 8: 325–334

    Article  MATH  Google Scholar 

  4. Abhyankar S. and Moh T. (1975). Embeddings of the line in the plane. J. Reine Angew. Math. 276: 148–166

    MATH  MathSciNet  Google Scholar 

  5. Berry T. (1997). Parametrization of algebraic space curves. J. Pure Appl. Algebra 117: 81–95

    Article  MathSciNet  Google Scholar 

  6. El Houari, H., El Kahoui, M.: Algorithms for recognizing coordinates in two variables over UFDs. In: Proceedings of ISSAC’2004, pp. 135–140. ACM Press (2004)

  7. El Kahoui M. (2005). D-resultant and subresultants. Proc. Am. Math. Soc. 133: 2193–2199

    Article  MATH  MathSciNet  Google Scholar 

  8. Essen, A.: Polynomial Automorphisms and the Jacobian Conjecture, vol. 190. Birkhäuser edn. Prog. Math. (2000)

  9. Essen A. and Yu J. (1997). The D-resultant, singularities and the degree of unfaithfulness. Proc. Am. Math. Soc. 125(3): 689–695

    Article  MATH  Google Scholar 

  10. Gutierrez J., Rubio R. and Schicho J. (2002). Polynomial parametrization of curves without affine singularities. Comput. Aided Geom. Des. 19: 223–234

    Article  MATH  MathSciNet  Google Scholar 

  11. Hoeij M. (1997). Rational parametrizations of algebraic curves using canonial divisors. J. Symb. Comput. 23: 209–227

    Article  MATH  Google Scholar 

  12. Lam, C., Shpilrain, V., Yu, J.: Recognizing and parametrizing curves isomorphic to a line. To appear in J. Symb. Comput. (2004)

  13. Manocha D. and Canny J. (1991). Rational curves with polynomial parametrization. Comput. Aided Des. 23(9): 645–652

    Article  MATH  Google Scholar 

  14. Recio T. and Sendra J. (1997). Real reparametrizations of real curves. J. Symb. Comput. 23(2–3): 241–254

    Article  MATH  MathSciNet  Google Scholar 

  15. Schicho J. (1992). On the choice of pencils in the parametrization of curves. J. Symb. Comput. 14(6): 557–576

    MATH  MathSciNet  Google Scholar 

  16. Sendra J. and Villarino C. (2001). Optimal reparametrization of polynomial algebraic curves. Int. J. Comput. Geom. Appl. 11(4): 439–453

    Article  MATH  MathSciNet  Google Scholar 

  17. Sendra J. and Winker F. (1991). Symbolic parametrization of curves. J. Symb. Comput. 12: 607–631

    MATH  Google Scholar 

  18. Sendra J. and Winkler F. (1991). Symbolic parametrization of curves. J. Symb. Comput. 12(6): 607–631

    Article  MATH  MathSciNet  Google Scholar 

  19. Sendra J. and Winkler F. (1997). Parametrization of algebraic curves over optimal field extension. J. Symb. Comput. 23(2–3): 191–208

    Article  MATH  MathSciNet  Google Scholar 

  20. Sendra J. and Winkler F. (2001). Tracing index of rational curve parametrizations. Comput. Aided Geom. Des. 18(8): 771–795

    Article  MATH  MathSciNet  Google Scholar 

  21. Shpilrain V. and Yu J. (1997). Polynomial automorphisms and Gröbner reductions. J. Algebra 197(2): 546–558

    Article  MATH  MathSciNet  Google Scholar 

  22. Wright D. (1981). On the Jacobian Conjecture. Illinois J. Math 25(3): 423–440

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, P. O. Box 2390, Marrakech, Morocco

    Toufiq Benbouziane, Hassan El Houari & M’hammed El Kahoui

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  1. Toufiq Benbouziane
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  2. Hassan El Houari
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  3. M’hammed El Kahoui
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Correspondence to M’hammed El Kahoui.

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Benbouziane, T., El Houari, H. & El Kahoui, M. Polynomial parametrization of nonsingular complete intersection curves. AAECC 18, 483–493 (2007). https://doi.org/10.1007/s00200-007-0050-0

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  • DOI: https://doi.org/10.1007/s00200-007-0050-0

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