Numerical Semigroups and Applications (RSME Springer Series, 3)

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. Preface Acknowledgments Contents 1 Numerical Semigroups, the Basics 1.1 Notable Elements 1.2 Numerical Semigroups with Maximal Embedding Dimension 1.3 Special Gaps and Unitary Extensions of a Numerical Semigroup 2 Irreducible Numerical Semigroups 2.1 Characterizations of Irreducible Numerical Semigroups 2.2 Decomposition of a Numerical Semigroup into Irreducible Semigroups 2.3 Free Numerical Semigroups 3 Ideals 3.1 Basic Definitions and Relevant Invariants 3.2 Duality 3.3 Irreducibility 3.4 Reduction Number, Blowup and Multiplicity Sequence; Arf Semigroups 4 Semigroup of an Irreducible Meromorphic Series 4.1 Some Notation 4.2 Characteristic Sequences 4.3 Contact 4.4 The Local Case 4.4.1 Module of Kähler Differentials 4.5 The Case of Curves with One Place at Infinity 4.5.1 Module of Kähler Differentials on Polynomial Curves 5 Minimal Presentations 5.1 Generators and Relations 5.2 Free Numerical Semigroups 5.3 Graphs of Factorizations and Minimal Presentations 5.4 Presentations and Binomial Ideals 5.5 Shaded Sets and Generating Functions 6 Factorizations and Divisibility 6.1 Length Based Invariants 6.2 Distance Based Invariants 6.3 How Far Is an Irreducible from Being Prime 6.4 Divisors and Feng–Rao Distances Appendix References Index