This book is an extended and revised version of "Numerical Semigroups with Applications," published by Springer as part of the RSME series. Like the first edition, it presents applications of numerical semigroups in Algebraic Geometry, Number Theory and Coding Theory. It starts by discussing the basic notions related to numerical semigroups and those needed to understand semigroups associated with irreducible meromorphic series. It then derives a series of applications in curves and factorization invariants. A new chapter is included, which offers a detailed review of ideals for numerical semigroups. Based on this new chapter, descriptions of the module of Kähler differentials for an algebroid curve and for a polynomial curve are provided. Moreover, the concept of tame degree has been included, and is viewed in relation to other factorization invariants appearing in the first edition. This content highlights new applications of numerical semigroups and their ideals, following in the spirit of the first edition.

Addressed to undergraduate students, and to reasearchers who want to see 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

Autorentext

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

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

Marco D'Anna obtained his PhD form the University of Roma La Sapienza in 1997 and became an Associated Professor at Catania University in 2001, where he has been supervisor of many PhD and master students. He has published several research papers on Commutative Algebra, mainly on one-dimensional rings and on numerical semigroup rings.

Inhalt

1 Numerical semigroups, the basics.- 2 Irreducible numerical semigroups.- 3 Ideals.- 4 Semigroup of an irreducible meromorphic series.- 5 Minimal presentations.- 6 Factorizations and divisibility.