A closed formula for the Geil-Matsumoto bound on numerical semigroups via Apéry sets
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2025
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| _version_ | 1866917536992329728 |
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| author | Marques, Adler Mendoza, Erik Quoos, Luciane Tizziotti, Guilherme |
| author_facet | Marques, Adler Mendoza, Erik Quoos, Luciane Tizziotti, Guilherme |
| contents | 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 $\lceil\textstyle\frac{n-1}{2}\rceil\leq t \leq n-1$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17022 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A closed formula for the Geil-Matsumoto bound on numerical semigroups via Apéry sets Marques, Adler Mendoza, Erik Quoos, Luciane Tizziotti, Guilherme Number Theory 11G20, 14G15, 14H05, 14Q05 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 $\lceil\textstyle\frac{n-1}{2}\rceil\leq t \leq n-1$, 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. |
| title | A closed formula for the Geil-Matsumoto bound on numerical semigroups via Apéry sets |
| topic | Number Theory 11G20, 14G15, 14H05, 14Q05 |
| url | https://arxiv.org/abs/2508.17022 |