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.
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
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 StanleyReisner Rings and Edge Ideals 7 Edge
Ideals of Graphs 8 Toric Ideals and Affine Varieties 9 Linear and ReedMuller
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
Rafael H. Villarreal is a Professor at Centro de Investigaci´on y de Estudios Avanzados del Instituto Polit´ecnico Nacional (Cinvestav), earned a Ph.D. degree from Rutgers University(1986). He has published 102 research papers in collaboration with 60 co-authors from various countries, has supervised 11 doctoral dissertations, and has systematically employed combinatorial and computational methods in commutative algebra, and its relation to toric ideals, polyhedral geometry, evaluation codes, combinatorial optimization, algebraic graph theory, graphs and clutters, and topics on monomial Cremona transformations.
