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Full Description
Monomial Algebras presents algebraic, combinatorial, and computational methods for studying monomial algebras and their ideals, including Stanley-Reisner rings, monomial subrings, Ehrhart rings, and blowup algebras. It emphasizes square-free monomials and the corresponding graphs, clutters, or hypergraphs.
New to the Third Edition
· Two full new chapters covering linear and Reed-Muller-type codes (chapter 9), and the containment problem and the resurgence of ideals (chapter 16)
· Extensive addition of new sections throughout the existing chapters to bring the book up-to-date and make it even more comprehensive
· A new appendix detailing software procedures for use alongside the book.
Bringing together several areas of pure and applied mathematics, this book shows how monomial algebras are related to polyhedral geometry, combinatorial optimization, and combinatorics of hypergraphs. It directly links the algebraic properties of monomial algebras to combinatorial structures (such as simplicial complexes, posets, digraphs, graphs, and clutters) and linear optimization problems.
Contents
Preface 1 Polyhedral Geometry and Linear Optimization 2 Commutative Algebra 3 Affine and Graded Algebras 4 Rees Algebras and Normality 5 Hilbert Series and Gorenstein Rings 6 Stanley-Reisner Rings and Edge Ideals 7 Edge Ideals of Graphs 8 Toric Ideals and Affine Varieties 9 Linear and Reed-Muller Type Codes 10 Monomial Subrings 11 Monomial Subrings of Graphs 12 Edge Subrings and Combinatorial Optimization 13 Normality of Rees Algebras of Monomial Ideals 14 Combinatorics of Symbolic Rees Algebras of Edge Ideals of Clutters 15 Combinatorial Optimization and Blowup Algebras 16 The Containment Problem and the Resurgence of Ideals A Procedures for Macaulay2 and Normaliz B Graph Diagrams Bibliography Notation Index Index
