标题： 半群和具有许多点的曲线的Geil-Matsumoto界的一个闭合公式 显示英文标题

标题： A closed formula for the Geil-Matsumoto bound on semigroups and curves with many points

摘要： 吉尔-松本界（GM界）限制了有限域上曲线的有理点数量，这取决于曲线上任意一点的魏尔斯特拉斯半群。 对于一般的数值半群，GM界缺乏一个简单的显式表达式，使其计算成为一个具有挑战性的问题。 当半群由两个互质整数生成时，已经得到了一个闭合公式。 在本工作中，对于任何数值半群，我们提供了以半群中非零元素的阿佩里集表示的GM界的闭合公式。 在数值半群由连续整数$n, n+1, \dots, n+t$生成且$\lceil\textstyle\frac{n-1}{2}\rceil\leq t \leq n-1$的情况下，我们得到了该界的简单闭合公式。 我们将这些结果应用于有限域上的代数曲线，以获得有理点数量的上界。 在某些情况下，我们的界改进了一些已知的有理点数量的上界。 显示英文摘要

摘要： The Geil-Matsumoto bound (GM bound) constrains the number of rational points on a curve over a finite field in terms of the Weierstrass semigroup of any of the points on the curve. For general numerical semigroups, the GM bound lacks a simple closed-form expression, making its computation a challenging problem. A closed formula has been obtained for the case when the semigroup is generated by two co-prime integers. In this work, for any numerical semigroup, we provide a closed formula for the GM bound in terms of the Ap\'ery set of a nonzero element of the semigroup. In the case where the numerical semigroup is generated by consecutive integers $n, n+1, \dots, n+t$ with $\lceil\textstyle\frac{n-1}{2}\rceil\leq t \leq n-1$, we obtain a simple closed formula for the bound. We apply these results to obtain upper bounds on the number of rational points for algebraic curves over finite fields. In some cases, our bounds improve some well-known upper bounds on the number of rational points.