Singular algebraic curves over finite fields
This paper presents algorithms for the identification and resolution of rational and non-rational singularities (by means of blowings-up) of a projective plane curve C: F(x1, x2, x3)=0 with coefficients in a finite field k. As a result, the genus of the curve is computed. In addition the running time of the algorithms are also analyzed.
Key words and phrasesfinite field rational points non-rational points singular points blowing-up process genus
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- 1.S. Abhyankar and C. Bajaj. Computations with algebraic curves. Lecture Notes in Computer Science, Springer-Verlag, 358, July 1988.Google Scholar
- 2.M. Bronstein, M. Hassner, A. Vasquez, and C.J. Williamson. Computer algebra algorithms for the construction of error correcting codes on algebraic curves. IEEE Proceedings on Information Theory, June 1991.Google Scholar
- 3.D. Le Brigand and J.J. Risler. Algorithm de Brill-Noether et codes de Goppa. Bull. Soc. math. France, 116:231–253, 1988.Google Scholar
- 4.D. Polemi. The Brill-Noether Theorem for Finite Fields and an Algorithm for Finding Algebraic Geometric Goppa Codes. Proceedings of the IEEE International Symposium on Information Theory, p.38, Budapest, Hungary, 1991.Google Scholar
- 5.D. Polemi, C. Moreno, and O. Moreno. A construction of a.g. Goppa codes from singular curves. submitted for publication, 1992.Google Scholar
- 6.T. Sakkalis. The topological configuration of a real algebraic curve. Bull. Austr. Math. Soc., 43:37–50, 1991.Google Scholar
- 7.T. Sakkalis and R. Farouki. Singular points of algebraic curves. Journal of Symbolic Computation, 9:405–421, 1990.Google Scholar
- 8.S. Vladut and Y. Manin. Linear codes and modular curves. Journal of Soviet Mathematics, 30, no. 6:2611–2643, 1985.Google Scholar