Abstract. We determine the Weierstrass semigroup H(P ∞ , P 1 , . . . , P m ) at several points on the GK curve. In addition, we present conditions to find pure gaps on the set of gaps G(P ∞ , P 1 , . . . , P m ). Finally, we apply the results to obtain AG codes with good relative parameters.
Abstract. We determine the Weierstrass semigroup H(P ∞ , P 1 , . . . , P m ) at several points on the GK curve. In addition, we present conditions to find pure gaps on the set of gaps G(P ∞ , P 1 , . . . , P m ). Finally, we apply the results to obtain AG codes with good relative parameters.
“…Matthews's approach was used to compute the generating set of classical Weierstrass semigroups at some collinear points on the Hermitian and Norm-trace curves in [16] and [18], respectively. A general study of the generating set of classical Weierstrass semigroups at several points on certain curves of the form f (y) = g(x), which include many classic curves over finite fields, was made in [5] by Castellanos and Tizziotti. In [7], Delgado proposed a different generalization of Weierstrass semigroups at a single point to several points on a curve. His original approach was on curves over algebraically closed fields, but later Beelen and Tutas [2] studied such objects for curves defined over finite fields, where they presented many properties of these semigroups.…”
In this work we study the generalized Weierstrass semigroup H(P m ) at an m-tuple P m = (P 1 , . . . , P m ) of rational points on certain curves admitting a plane model of the form fIn particular, we compute the generating set Γ(P m ) of H(P m ) and, as a consequence, we explicit a basis for Riemann-Roch spaces of divisors with support in {P 1 , . . . , P m } on these curves, generalizing results of Maharaj, Matthews, and Pirsic in [14].
“…Este trabalho pretende sintetizar alguns dos resultados atualmente conhecidos dos semigrupo de Weierstrass sobre vários pontos. Especificamente o trabalho de Carvalho e Torres em [4], os resultados de Matthews em [21] e Castellanos e Tizziotti em [5].…”
Section: Introductionunclassified
“…Em [21], Matthews estabelece o semigrupo de Weierstrass em m pontos colineares para uma curva hermitiana, este resultado foi obtido a partir da noção de conjunto gerador mínimo para o semigrupo de Weierstrass, introduzido por ela. Em [5], Castellanos e Tizziotti, usam o conceito de discrepância para caracterizar os elementos mínimos de um semigrupo de Weierstrass para uma família de curvas com variáveis separáveis com o modelo afim f pyq " gpxq.…”
Section: Introductionunclassified
“…O capítulo 3é dedicado ao cálculo do semigrupo de Weierstrass em dois e vários pontos na curva Hermitiana [21] e [20]. Finalmente, no capítulo 4 vamos expor o trabalho de Castellanos e Tizziotti em [5]. Na seção 4.1, introduzimos o conceito de discrepância e sua relação com os elementos mínimos do semigrupo Weierstrass.…”
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