Numerical Semigroups : IMNS 2018

"This book presents the state of the art on numerical semigroups and related subjects, offering different perspectives on research in the field and including results and examples that are very difficult to find in a structured exposition elsewhere. The contents comprise the proceedings of the 2018 INdAM "International Meeting on Numerical Semigroups", held in Cortona, Italy. Talks at the meeting centered not only on traditional types of numerical semigroups, such as Arf or symmetric, and their usual properties, but also on related types of semigroups, such as affine, Puiseux, Weierstrass, and primary, and their applications in other branches of algebra, including semigroup rings, coding theory, star operations, and Hilbert functions. The papers in the book reflect the variety of the talks and derive from research areas including Semigroup Theory, Factorization Theory, Algebraic Geometry, Combinatorics, Commutative Algebra, Coding Theory, and Number Theory. The book is intended for researchers and students who want to learn about recent developments in the theory of numerical semigroups and its connections with other research fields."--Page [4] of cover Preface 6 Contents 8 Counting Numerical Semigroups by Genus and Even Gaps via Kunz-Coordinate Vectors 10 1 Introduction 10 2 Apéry Set and Kunz-Coordinate Vector 12 3 The Main Result and an Application to a Counting Problem 13 References 17 Patterns on the Numerical Duplication by Their Admissibility Degree 18 1 Introduction 18 2 Preliminaries 19 3 Patterns and Their Admissibility Degree 21 4 Patterns Equivalent to the Arf Pattern 24 5 Patterns on the Numerical Duplication 27 6 Patterns on Rings 32 References 34 Primality in Semigroup Rings 35 1 Introduction 35 2 Primal Elements in Monoids 36 3 Primal Elements in a Graded Domain 39 4 Primal Elements in Semigroup Rings 43 References 46 Conjecture of Wilf: A Survey 47 1 Introduction 47 1.1 Terminology and Notation 48 1.2 A Convenient Way to Visualize Numerical Semigroups 49 2 Two Problems Posed by Wilf 52 2.1 Wilf's Paper 52 2.2 Problem (a.i): Wilf's Conjecture 53 2.3 Problem (a.ii): Another Open Problem 53 2.4 Problem (b): Counting Numerical Semigroups 54 3 Some Classes of Wilf Semigroups 55 3.1 The Type as an Important Ingredient 56 3.2 Semigroups Given by Sets of Generators 58 3.3 Semigroups with Nonnegative Eliahou Numbers 59 3.4 Natural Constructions 60 3.5 Semigroups with Small Multiplicity 60 3.6 Semigroups with Large Embedding Dimension (Compared to the Multiplicity) 61 3.6.1 Some Comments 61 3.7 Semigroups with Big Multiplicity (and Possibly Small Embedding Dimension) 62 3.8 Semigroups with Large Multiplicity (Compared to the Conductor) 62 3.8.1 Some Comments 63 3.9 Considering Unusual Invariants 63 3.10 Families Described Through One Invariant 64 4 Quasi-Generalization 64 References 69 Gapsets of Small Multiplicity 71 1 Introduction 71 2 Gapset Filtrations 73 2.1 The Canonical Partition 74 2.2 Gapset Filtrations 75 3 The Case m=3 77 4 Some More General Tools 80 4.1 On m-Extensions and m-Filtrations 81 4.2 Gapset Filtrations Revisited 82 4.3 A Compact Representation 82 4.4 Complementing an m-Extension 84 4.5 The Insertion Maps fi 87 5 The Case m=4 87 5.1 Concluding Remark 89 References 89 Generic Toric Ideals and Row-Factorization Matrices in Numerical Semigroups 91 1 Preliminaries 91 1.1 Numerical Semigroups 91 1.2 The Fibers of Elements in Numerical Semigroups 92 1.3 Semigroup Rings 95 2 Generic Toric Ideals 96 2.1 Main Results 96 2.2 Basic Fibers 97 References 99 Symmetric (Not Complete Intersection) Semigroups Generated by Six Elements 100 1 Introduction 100 2 Symmetric (Not CI) Semigroups Generated by Six Integers and Polynomial Identities 101 3 The Lower Bound for the Frobenius Numbers of Semigroups S6 106 4 Are There Any Constraints on Betti's Numbers of Symmetric (Not CI) Semigroups S6? 110 5 Symmetric (Not CI) Semigroups S6 with the W and W2 Properties 112 References 115 Syzygies of Numerical Semigroup Rings, a Survey Through Examples 117 1 Introduction 117 2 Principal Matrices and Gorenstein Sequences of Length 4 119 3 Arithmetic Sequences 125 4 Gluing 129 References 133 Irreducibility and Factorizations in Monoid Rings 134 1 Introduction 134 2 Notation and Background 135 2.1 General Notation 135 2.2 Monoids 136 2.3 Factorizations 136 2.4 Monoid Rings 137 3 Irreducibility Criteria for Monoid Rings 138 3.1 Extended Gauss's Lemma 138 3.2 Extended Eisenstein's Criterion 139 4 Factorizations in Monoid Algebras 141 References 143 On the Molecules of Numerical Semigroups, Puiseux Monoids, and Puiseux Algebras 145 1 Introduction 145 2 Monoids, Atoms, and Molecules 147 2.1 General Notation 147 2.2 Monoids 148 2.3 Atoms and Molecules 148 3 Molecules of Numerical Semigroups 149 3.1 Betty Elements 150 3.2 On the Sizes of the Sets of Molecules 151 4 Molecules of Puiseux Monoids 152 4.1 Molecules of Generic Puiseux Monoids 152 4.2 Molecules of Prime Reciprocal Monoids 155 5 Molecules of Puiseux Algebras 160 References 164 Arf Numerical Semigroups with Multiplicity 9 and 10 166 1 Introduction 166 2 Arf Numerical Semigroups 167 3 Arf Numerical Semigroups with Multiplicity 9 174 4 Arf Numerical Semigroups with Multiplicity 10 179 References 186 Numerical Semigroup Rings of Maximal Embedding Dimension with Determinantal Defining Ideals 187 1 Introduction 187 2 Proof of Theorem 2 190 2.1 Proof of (3) (2) and the Assertion (a) 190 2.2 Proof of (2) (4) 193 2.3 Proof of (b) and (c) 195 3 Examples 197 References 198 Embedding Dimension of a Good Semigroup 199 1 Introduction 199 2 Semiring Associated to a Good Semigroup and Irreducible Absolutes 201 2.1 Semiring ΓS and Basic Properties 201 2.2 A System of Generators of ΓS as a Semiring 204 3 Embedding Dimension of a Good Semigroup 207 3.1 An Inferior Bound for the Embedding Dimension 212 3.2 A Superior Bound for the Embedding Dimension 215 3.3 An Algorithm for the Computation of the Embedding Dimension of a Semigroup SN2 219 4 Properties of Embedding Dimension 223 4.1 Relationship Between Embedding Dimension of a Ring and Embedding Dimension of Its Value Semigroup 223 4.2 Relationship Between Embedding Dimension and Multiplicity 226 References 232 On Multi-Index Filtrations Associated to Weierstraß Semigroups 233 1 Introduction 233 2 Terminology and Notation 234 2.1 Branches and Parametrizations 235 2.2 Divisors on Smooth Curves 236 3 Brill-Noether Theory for Curves 237 4 The Weierstraß Semigroup at Several Points 242 4.1 Dimension of the Riemann-Roch Quotients with Respect to Pi and Associated Functions 244 4.2 Computing the Weierstraß Semigroup at Two Points 250 5 Computational Aspects Using Singular 254 5.1 Hints of Usage of brnoeth.lib 254 5.2 Procedures Generalizing to Several Points 255 References 259 On the Hilbert Function of Four-Generated Numerical SemigroupRings 261 1 Introduction 261 2 Preliminaries 262 3 The Case Embedding Dimension 4 266 4 Cases with Cardinality of A2≤ 4 268 5 Case D2 = {n3, n4} 274 6 The Hilbert Function Does Not Decrease at Level 3 278 7 Conclusions 283 References 285 Lattice Ideals, Semigroups and Toric Codes 287 1 Introduction 287 2 The Algebraic Approach 289 3 Lattice Ideals and Subgroups of the Torus TX 296 3.1 Degenerate Tori 297 4 Vanishing Ideals of Subsgroups of TX 298 4.1 The Length of the Code Cα,YQ 301 References 303 The Number of Star Operations on Numerical Semigroups and on Related Integral Domains 305 1 Introduction 305 2 Notation 306 3 Star Operations 307 4 Estimates Through the Genus 308 5 Estimates Through the Multiplicity 309 6 Multiplicity 3 311 7 Prime Multiplicity 315 8 Linear Families 319 9 Algorithms and Explicit Data 321 10 The Ring Version 324 References 328 Torsion in Tensor Products over One-Dimensional Domains 330 1 Introduction 330 2 Some Evidence 331 References 334 Almost Symmetric Numerical Semigroups with Odd Generators 336 1 Introduction 336 2 Basic Concepts 338 3 Symmetric Semigroups 339 4 Pseudo-Symmetric Semigroups 341 5 Almost Symmetric Semigroups with Type Three 343 References 349 Poincaré Series on Good Semigroup Ideals 351 1 Introduction 351 2 Preliminaries 352 2.1 Good Semigroups and Their Ideals 353 2.2 Distance of Semigroup Ideals 354 2.3 Canonical Semigroup Ideals 355 2.4 Value Semigroups 356 3 Distance and Duality 358 4 Symmetry of the Poincaré Series 362 References 367 A Short Proof of Bresinski's Theorem on Gorenstein Semigroup Rings Generated by Four Elements 368 1 Basic Concepts 368 2 The Proof 370 References 372