Generalized Weierstrass semigroups and their Poincaré series

TL;DR

The paper addresses describing generalized Weierstraß semigroups $\widehat{H}(\mathbf Q)$ at multiple rational points on curves over finite fields and links their structure to Riemann–Roch spaces via a finite generating framework. It shows that $\widehat{H}(\mathbf Q)$ is completely determined by its absolute maximal elements and that the semigroup can be recovered as the least upper bound of such maxima, enabling finitely many generators. The authors extend the Poincaré-series approach to the multivariate case, proving that $P(\mathbf t)$ encodes the entire semigroup and factors through a semigroup polynomial $P^*(\mathbf t)$, with symmetry yielding multivariate functional equations. These results connect the semigroup structure to divisors supported on the chosen points and to RR-spaces, providing computational tools with potential applications to algebraic-geometry codes and divisor theory.

Abstract

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 deduce properties concerned with the arithmetical structure of divisors supported on the specified points and their corresponding Riemann-Roch spaces. This characterization allows us to show that the Poincaré series associated with generalized Weierstrass semigroups carry essential information to describe entirely their respective semigroups.

Figures (1)

• Figure 1: Maximal elements of $\widehat{H}(Q,P)$ in the Example \ref{['ex']}

Theorems & Definitions (46)

• Proposition 2.1
• proof
• Remark 2.2
• Corollary 2.3
• Definition 2.4
• Remark 2.5
• Lemma 2.6
• proof
• Definition 2.7
• Proposition 2.8