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Proportionally modular diophantine inequalities and their multiplicity

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Abstract

Let I be an interval of positive rational numbers. Then the set S (I) = T ∩ ℕ, where T is the submonoid of ℚ +0 , +) generated by T, is a numerical semigroup. These numerical semigroups are called proportionally modular and can be characterized as the set of integer solutions of a Diophantine inequality of the form ax mod b ≤ cx. In this paper we are interested in the study of the maximal intervals I subject to the condition that S (I) has a given multiplicity. We also characterize the numerical semigroups associated with these maximal intervals.

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Authors and Affiliations

  1. Departamento de Álgebra, Universidad de Granada, E-18071, Granada, Spain

    José Carlos Rosales

  2. Departamento de Matemática, Évora, 7000, Évora, Portugal

    Manuel Batista Branco

  3. Departamento de Matemática, Universidade de Trás-os-Montes e Alto Douro, 5001-801, Vila Real, Portugal

    Paulo Vasco

Authors
  1. José Carlos Rosales
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  2. Manuel Batista Branco
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  3. Paulo Vasco
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Corresponding author

Correspondence to José Carlos Rosales.

Additional information

The first author is supported by the project MTM2004-01446 and FEDER funds; the paper is supported by the Luso-Espanhola action HP2004-0056

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Rosales, J.C., Branco, M.B. & Vasco, P. Proportionally modular diophantine inequalities and their multiplicity. Acta. Math. Sin.-English Ser. 26, 2059–2070 (2010). https://doi.org/10.1007/s10114-010-7573-1

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  • DOI: https://doi.org/10.1007/s10114-010-7573-1

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