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On Weierstrass semigroups and sets: a review with new results

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Abstract

In this work we present a survey of the main results in the theory of Weierstrass semigroups at several points, with special attention to the determination of bounds for the cardinality of its set of gaps. We also review results on applications to the theory of error correcting codes. We then recall a generalization of the concept of Weierstrass semigroup, which is the Weierstrass set associated to a linear system and several points. We finish by presenting new results on this Weierstrass set, including some on the cardinality of its set of gaps.

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References

  1. Arbarello E., Cornalba M., Griffiths P., Harris J.: Geometry of Algebraic Curves. Springer- Verlag, Berlin (1985)

    MATH  Google Scholar 

  2. Ballico E.: Weierstrass points and Weierstrass pairs on algebraic curves. Int. J. Pure Appl. Math. 2, 427–440 (2002)

    MATH  MathSciNet  Google Scholar 

  3. Ballico E., Kim S.J.: Weierstrass multiple loci of n-pointed algebraic curves. J. Algebra 199, 455–471 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Beelen P., Tutaş N.: A generalization of the Weierstrass semigroup. J. Pure Appl. Algebra 207, 243–260 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Carvalho C.: On V-Weierstrass sets and gaps. J. Algebra 312, 956–962 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Carvalho C., Kato T.: Codes from curves with total inflection points. Des. Codes Cryptogr. 45, 359–364 (2007)

    Article  MathSciNet  Google Scholar 

  7. Carvalho, C., Silva, E.: On algebras admitting a complete set of near weights, evaluation codes and Goppa codes. preprint arXiv:0807.3198 (July 2008)

  8. Carvalho C., Torres F.: On Goppa codes and Weierstrass gaps at several points. Des. Codes Cryptogr. 35, 211–225 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Garcia, A., Lax, R.F.: Goppa codes and Weierstrass gaps. In: Coding Theory and Algebraic Geometry. Proc. Int. Workshop, Luminy/Fr. 1991, Lecture Notes in Mathematics, vol. 1518, pp. 33–42 (1992)

  10. Garcia A., Kim S.J., Lax R.F.: Consecutive Weierstrass gaps and minimum distance of Goppa codes. J. Pure Appl. Algebra 84, 199–207 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  11. Goldschmidt D.M.: Algebraic Functions and Projective Curves. Graduate Texts in Mathematics, Vol. 215. Springer-Verlag, New York (2003)

    Google Scholar 

  12. Homma M.: The Weierstrass semigroup of a pair of points on a curve. Arch. Math. 67, 337–348 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Homma M., Kim S.J.: Goppa codes with Weierstrass pairs. J. Pure Appl. Algebra 162, 273–290 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Homma M., Kim S.J., Komeda J.: A semigroup at a pair of Weierstrass points on a cyclic 4-gonal curve and a bielliptic curve. J. Algebra 305, 1–17 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ishii N.: A certain graph obtained from a set of several points on a Riemann surface. Tsukuba J. Math. 23, 55–89 (1999)

    MATH  MathSciNet  Google Scholar 

  16. Ishii N.: Weierstrass gap sets for quadruples of points on compact Riemann surfaces. J. Algebra 250, 44–66 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kang E., Kim S.J.: Special pairs in the generating subset of the Weierstrass semigroup at a pair. Geom. Dedicata 99, 167–177 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kim S.J.: On the index of the Weierstrass semigroup of a pair of points on a curve. Arch. Math. 62, 73–82 (1994)

    Article  MATH  Google Scholar 

  19. Kim S.J., Komeda J.: The Weierstrass semigroup of a pair and moduli in \({\mathcal {M}_3}\) . Bol. Soc. Brasil. Mat. Nova. Ser. 32, 149–157 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kim S.J., Komeda J.: Weierstrass semigroups of a pair of points whose first nongaps are three. Geom. Dedicata 93, 113–119 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Matthews G.L.: Weierstrass pairs and minimum distance of Goppa codes. Des. Codes Cryptogr. 22, 107–121 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  22. Matthews, G.L.: The Weierstrass Semigroup of an m-Tuple of Collinear Points on a Hermitian Curve. Finite Fields and Applications. Lecture Notes in Computer Science, vol. 2948, pp. 12–24. Springer, Berlin (2004)

  23. Matthews G.L.: Codes from the Suzuki function field. IEEE Trans. Inf. Theory 50, 3298–3302 (2004)

    Article  MathSciNet  Google Scholar 

  24. Matthews G.L.: Weierstrass semigroups and codes from a quotient of the Hermitian curve. Des. Codes Cryptogr. 37, 473–492 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Stichtenoth H.: Algebraic Function Fields and Codes. Springer-Verlag, Berlin (1993)

    MATH  Google Scholar 

  26. Stöhr K.-O., Voloch J.F.: Weierstrass points and curves over finite fields. Proc. Lond. Math. Soc. (III. Ser.) 52, 1–19 (1986)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculdade de Matemática, Universidade Federal de Uberlândia, Av. J. N. Ávila 2121, 38.408-902, Uberlândia, MG, Brazil

    Cícero Carvalho

  2. Department of Mathematical Sciences, Faculty of Science, Yamaguchi University, Yamaguchi, 753-8512, Japan

    Takao Kato

Authors
  1. Cícero Carvalho
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  2. Takao Kato
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Correspondence to Cícero Carvalho.

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Carvalho, C., Kato, T. On Weierstrass semigroups and sets: a review with new results. Geom Dedicata 139, 195–210 (2009). https://doi.org/10.1007/s10711-008-9337-y

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  • DOI: https://doi.org/10.1007/s10711-008-9337-y

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