Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

REMARKS ON THE GAP SET OF R = K + C

  • Published : 2010.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

$\tilde{G}(P,\;Q)$, a new generalization of the set of gap numbers of a pair of points, was described in [1]. Here we study gap numbers of local subring $R\;=\;\cal{K}\;+\;C$ of algebraic function field over a finite field and we give a formula for the number of elements of $\tilde{G}(P,\;Q)$ depending on pure gaps and R.

Keywords

References

  1. P. Beelen and N. Tutas, A generalization of the Weierstrass semigroup, J. Pure Appl. Algebra 207 (2006), no. 2, 243–260. https://doi.org/10.1016/j.jpaa.2005.09.017
  2. A. Garcia and P. Viana, Weierstrass points on certain nonclassical curves, Arch. Math. (Basel) 46 (1986), no. 4, 315–322. https://doi.org/10.1007/BF01200462
  3. M. Homma, The Weierstrass semigroup of a pair of points on a curve, Arch. Math. (Basel) 67 (1996), no. 4, 337–348. https://doi.org/10.1007/BF01197599
  4. E. Kang and S. J. Kim, Special pairs in the generating subset of the Weierstrass semigroup at a pair, Geom. Dedicata 99 (2003), 167–177. https://doi.org/10.1023/A:1024960704513
  5. H. I. Karakas, On Rosenlicht's generalization of Riemann-Roch theorem and generalized Weierstrass points, Arch. Math. (Basel) 27 (1976), no. 2, 134–147. https://doi.org/10.1007/BF01224653
  6. H. I. Karakas, Application of generalized Weierstrass points: divisibility of divisor classes, J. Reine Angew. Math. 299/300 (1978), 388–395.
  7. S. J. Kim, On the index of the Weierstrass semigroup of a pair of points on a curve, Arch. Math. (Basel) 62 (1994), no. 1, 73–82. https://doi.org/10.1007/BF01200442
  8. G. Matthews, Weierstrass pairs and minimum distance of Goppa codes, Des. Codes Cryptogr. 22 (2001), no. 2, 107–121. https://doi.org/10.1023/A:1008311518095
  9. G. Matthews, Codes from the Suzuki function field, IEEE Trans. Inform. Theory 50 (2004), no. 12, 3298–3302. https://doi.org/10.1109/TIT.2004.838102
  10. M. Rosenlicht, Equivalence relations on algebraic curves, Ann. of Math. (2) 56 (1952), 169–191. https://doi.org/10.2307/1969773
  11. H. Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, Berlin, 1993.
  12. N. Tuta¸s, On Weierstrass point of semilocal subrings, JP J. Algebra Number Theory Appl. 13 (2009), no. 2, 221–235.