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Integral points in rational polygons: a numerical semigroup approach

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Abstract

In this paper we use an elementary approach by using numerical semigroups (specifically, those with two generators) to give a formula for the number of integral points inside a right-angled triangle with rational vertices. This is the basic case for computing the number of integral points inside a rational (not necessarily convex) polygon.

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Notes

  1. Throughout this paper we will call a point \(P \in {\mathbb {R}}^2\) integral if its coordinates lie in \({\mathbb {Z}}^2\), and similarly P will be called rational if \(P \in {\mathbb {Q}}^2\).

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Acknowledgments

The authors warmly acknowledge the help and advice of J. C. Rosales.

The final stage of this work was completed during a stay of the first author in the Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, during the spring term of 2013, thanks to the Grant P08–FQM–3894 (Junta de Andalucía).

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Authors and Affiliations

  1. Departamento de Álgebra and IMUS, Universidad de Sevilla, P.O. 1160, 41080, Seville, Spain

    Guadalupe Márquez-Campos & José M. Tornero

  2. Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier, Montpellier, France

    Jorge L. Ramírez-Alfonsín

Authors
  1. Guadalupe Márquez-Campos
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  2. Jorge L. Ramírez-Alfonsín
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  3. José M. Tornero
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Corresponding author

Correspondence to Guadalupe Márquez-Campos.

Additional information

Communicated by Fernando Torres.

Guadalupe Márquez-Campos and José M. Tornero were partially supported by the Grant FQM–218 and P12–FQM–2696 (FEDER and FSE). The Jorge L. Ramírez-Alfonsín was supported by ANR TEOMATRO Grant ANR-10-BLAN 0207 and Grant ECOS-Nord M13M01.

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Márquez-Campos, G., Ramírez-Alfonsín, J.L. & Tornero, J.M. Integral points in rational polygons: a numerical semigroup approach. Semigroup Forum 94, 123–138 (2017). https://doi.org/10.1007/s00233-016-9820-y

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  • DOI: https://doi.org/10.1007/s00233-016-9820-y

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