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© 2016

Numerical Semigroups and Applications

  • 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

Book

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

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Abdallah Assi, Pedro A. García-Sánchez
    Pages 1-15
  3. Abdallah Assi, Pedro A. García-Sánchez
    Pages 17-29
  4. Abdallah Assi, Pedro A. García-Sánchez
    Pages 31-68
  5. Abdallah Assi, Pedro A. García-Sánchez
    Pages 69-83
  6. Abdallah Assi, Pedro A. García-Sánchez
    Pages 85-99
  7. Back Matter
    Pages 101-106

About this book

Introduction

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.

Keywords

Numerical semigroup Algebraic curve AG codes Nonunique factorization invariants Combinatorics

Authors and affiliations

  1. 1.Dèpartement de MathématiquesUniversité d'AngersAngersFrance
  2. 2.Departamento de ÁlgebraUniversidad de GranadaGranadaSpain

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.


Bibliographic information

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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)​