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Numerical Semigroups and Applications

  • Book
  • © 2016

Overview

Authors:
  1. Abdallah Assi

    1. Dèpartement de Mathématiques, Université d'Angers, Angers, France

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    You can also search for this author in PubMed   Google Scholar

  2. Pedro A. García-Sánchez

    1. Departamento de Álgebra, Universidad de Granada, Granada, Spain

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • Useful for any undergraduate student, and also for researchers wishing to focus on the state of art in numerical semigroups research
  • Contains many examples and tutorials with the (free) numericalsgps GAP package
  • Shows the ubiquity of numerical semigroups
  • Includes supplementary material: sn.pub/extras

Part of the book series: RSME Springer Series (RSME, volume 1)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-xiv
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  2. Numerical Semigroups, the Basics

    • Abdallah Assi, Pedro A. García-Sánchez
    Pages 1-15

  3. Irreducible Numerical Semigroups

    • Abdallah Assi, Pedro A. García-Sánchez
    Pages 17-29

  4. Semigroup of an Irreducible Meromorphic Series

    • Abdallah Assi, Pedro A. García-Sánchez
    Pages 31-68

  5. Minimal Presentations

    • Abdallah Assi, Pedro A. García-Sánchez
    Pages 69-83

  6. Factorizations and Divisibility

    • Abdallah Assi, Pedro A. García-Sánchez
    Pages 85-99

  7. Back Matter

    Pages 101-106
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Keywords

About this book

This work presents applications of numerical semigroups in Algebraic Geometry, Number Theory, and Coding Theory. Background on numerical semigroups is presented in the first two chapters, which introduce basic notation and fundamental concepts and irreducible numerical semigroups. The focus is in particular on free semigroups, which are irreducible; semigroups associated with planar curves are of this kind. The authors also introduce semigroups associated with irreducible meromorphic series, and show how these are used in order to present the properties of planar curves. Invariants of non-unique factorizations for numerical semigroups are also studied. These invariants are computationally accessible in this setting, and thus this monograph can be used as an introduction to Factorization Theory. Since factorizations and divisibility are strongly connected, the authors show some applications to AG Codes in the final section. The book will be of value for undergraduate students(especially those at a higher level) and also for researchers wishing to focus on the state of art in numerical semigroups research.

Reviews

“This book gives some basic notions related to numerical semigroups, and presents applications of numerical semigroups in algebraic geometry, number theory and coding theory. … The book contains numerous examples and computational experiments. … This book will be of value for undergraduate students and also for researchers interested in numerical semigroups and their applications.” (Attila Nagy, Mathematical Reviews, May, 2017)​

Authors and Affiliations

  • Dèpartement de Mathématiques, Université d'Angers, Angers, France

    Abdallah Assi

  • Departamento de Álgebra, Universidad de Granada, Granada, Spain

    Pedro A. García-Sánchez

About the authors

Abdallah Assi graduated in Mathematics at the University Joseph Fourier (Grenoble, France). He obtained his Ph.D. in Mathematics at the same university and his HDR-Habilitation à diriger les recherches- at the University of Angers (France). He has a parmanent position at the Department of Mathematics in the University of Angers since 1995. His research interests are in affine geometry, numerical semigroups, and the theory of singularities.

Pedro A. Garcia-Sanchez was born in Granada, Spain, in 1969. Since 1992 he teaches in the Departmento de Algebra at the Universidad de Granada. He graduated in Mathematics and in Computer Science (Diploma) in 1992. He defended his PhD Thesis "Affine semigroups" in 1996, and since 1999 he has a permanent position at the Universidad de Granada. His main research interests are numerical semigroups, commutative monoids and nonunique factorization invariants.


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