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A closed formula for the Geil-Matsumoto bound on numerical semigroups via Apéry sets
Marques, Adler1; Mendoza, Erik1; Quoos, Luciane2; Tizziotti, Guilherme3
1Universidade Federal do Rio de Janeiro, Instituto de Matemática, Cidade Universitária, CEP 21941-909, Rio de Janeiro, Brazil
2Instituto de Matemática, Universidade Federal do Rio de Janeiro, Cidade Universitária, CEP 21941-909, Rio de Janeiro, Brazil
3Instituto de Matemática e Estatística, Universidade Federal de Uberlândia, Campus Santa Mônica, CEP 38400-902, Uberlândia, Brazil
Algebraic Combinatorics, Volume 9 (2026) no. 4, pp. 1139-1155

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 Apéry 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 $\lceil \tfrac{n-1}{2}\rceil \le t \le n-1$, 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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.503
Classification: 11G20, 14G15, 14H05, 14Q05
Keywords: Geil-Matsumoto bound, algebraic curves, rational points, Lewittes’ bound

Marques, Adler  1 ; Mendoza, Erik  1 ; Quoos, Luciane  2 ; Tizziotti, Guilherme  3

1 Universidade Federal do Rio de Janeiro, Instituto de Matemática, Cidade Universitária, CEP 21941-909, Rio de Janeiro, Brazil
2 Instituto de Matemática, Universidade Federal do Rio de Janeiro, Cidade Universitária, CEP 21941-909, Rio de Janeiro, Brazil
3 Instituto de Matemática e Estatística, Universidade Federal de Uberlândia, Campus Santa Mônica, CEP 38400-902, Uberlândia, Brazil
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Marques, Adler; Mendoza, Erik; Quoos, Luciane; Tizziotti, Guilherme. A closed formula for the Geil-Matsumoto bound on numerical semigroups via Apéry sets. Algebraic Combinatorics, Volume 9 (2026) no. 4, pp. 1139-1155. doi: 10.5802/alco.503
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