Assi A., D'Anna M., García-Sánchez P.A. Numerical Semigroups and Applications

2nd edition. — Springer Nature Switzerland AG, 2020. — 145 p. — ISBN: 978-3-030-54943-5 (eBook)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.Numerical Semigroups, the Basics
Notable Elements
Numerical Semigroups with Maximal Embedding Dimension
Special Gaps and Unitary Extensions of a Numerical Semigroup
Irreducible Numerical Semigroups
Characterizations of Irreducible Numerical Semigroups
Decomposition of a Numerical Semigroup into Irreducible Semigroups
Free Numerical Semigroups
Ideals
Basic Definitions and Relevant Invariants
Duality
Irreducibility
Reduction Number, Blowup and Multiplicity Sequence; Arf Semigroups
Semigroup of an Irreducible Meromorphic Series
Some Notation
Characteristic Sequences
Contact
The Local Case
Module of Kähler Differentials
The Case of Curves with One Place at Infinity
Module of Kähler Differentials on Polynomial Curves
Minimal Presentations
Generators and Relations
Free Numerical Semigroups
Graphs of Factorizations and Minimal Presentations
Presentations and Binomial Ideals
Shaded Sets and Generating Functions
Factorizations and Divisibility
Length Based Invariants
Distance Based Invariants
How Far Is an Irreducible from Being Prime
Divisors and Feng–Rao Distances
References
Index