Generalized Weierstrass semigroups and their Poincare series

We investigate the structure of the generalized Weierstrass semigroups at several points on a curve defined over a finite field. We present a description of these semigroups that enables us to associate them with combinatorial objects, the Poincare series and the semigroup polynomial. We show that this Poincare series determines completely the generalized Weierstrass semigroup and it is entirely determined by the semigroup polynomial. We finish the paper by describing the functional equations occurring to the Poincare series under the hypothesis of a symmetric generalized Weierstrass semigroup. (C) 2019 Elsevier Inc. All rights reserved.