2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/222866Let Lambda be a numerical semigroup. Assume there exists an algebraic function field over GF(q) in one variable which possesses a rational place that has Lambda as its Weierstrass semigroup. We ask the question as to how many rational places such a function field can possibly have and we derive an upper bound in terms of the generators of Lambda and q. Our bound is an improvement to a bound by Lewittes which takes into account only the multiplicity of Lambda and q. From the new bound we derive significant improvements to Serre's upper bound in the cases q=2, 3 and 4. We finally show that Lewittes' bound has important implications to the theory of towers of function fields.16 pages, 3 tablesAlgebraic GeometryNumber Theory14G15 (Primary); 11G20, 14H05, 14H25 (Secondary)text