Content: Front Cover; Contents; Preface; Preface to First Edition; 1. Polyhedral Geometry and Linear Optimization; 2. Commutative Algebra; 3. Affine and Graded Algebras; 4. Rees Algebras and Normality; 5. Hilbert Series; 6. Stanley-Reisner Rings and Edge Ideals of Clutters; 7. Edge Ideals of Graphs; 8. Toric Ideals and Affine Varieties; 9. Monomial Subrings; 10. Monomial Subrings of Graphs; 11. Edge Subrings and Combinatorial Optimization; 12. Normality of Rees Algebras of Monomial Ideals; 13. Combinatorics of Symbolic Rees Algebras of Edge Ideals of Clutters 14. Combinatorial Optimization and Blowup AlgebrasAppendix Graph Diagrams; Bibliography
Abstract:
The book stresses the interplay between several areas of pure and applied mathematics, emphasizing the central role of monomial algebras. It unifies the classical results of commutative algebra with central results and notions from graph theory, combinatorics, linear algebra, integer programming, and combinatorial optimization. The book introduces various methods to study monomial algebras and their presentation ideals, including Stanley-Reisner rings, subrings and blowup algebra-emphasizing square free quadratics, hypergraph clutters, and effective computational methods.