Author | Stanley, Richard P. author |
---|---|

Title | Combinatorics and Commutative Algebra [electronic resource] / by Richard P. Stanley |

Imprint | Boston, MA : Birkhรคuser Boston, 1996 |

Edition | Second Edition |

Connect to | http://dx.doi.org/10.1007/b139094 |

Descript | XII, 168 p. online resource |

SUMMARY

Some remarkable connections between commutative algebra and combinatorics have been discovered in recent years. This book provides an overview of two of the main topics in this area. The first concerns the solutions of linear equations in nonnegative integers. Applications are given to the enumeration of integer stochastic matrices (or magic squares), the volume of polytopes, combinatorial reciprocity theorems, and related results. The second topic deals with the face ring of a simplicial complex, and includes a proof of the Upper Bound Conjecture for Spheres. An introductory chapter giving background information in algebra, combinatorics and topology broadens access to this material for non-specialists. New to this edition is a chapter surveying more recent work related to face rings, focusing on applications to f-vectors. Included in this chapter is an outline of the proof of McMullen's g-conjecture for simplicial polytopes based on toric varieties, as well as a discussion of the face rings of such special classes of simplicial complexes as shellable complexes, matroid complexes, level complexes, doubly Cohen-Macaulay complexes, balanced complexes, order complexes, flag complexes, relative complexes, and complexes with group actions. Also included is information on subcomplexes and subdivisions of simplicial complexes, and an application to spline theory

CONTENT

Background -- Nonnegative Integral Solutions to Linear Equations -- The Face Ring of a Simplicial Complex -- Further Aspects of Face Rings

Mathematics
Commutative algebra
Commutative rings
Topology
Combinatorics
Mathematics
Combinatorics
Commutative Rings and Algebras
Topology