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© 1996

Combinatorics and Commutative Algebra

Textbook

Part of the Progress in Mathematics book series (PM, volume 41)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Pages 1-24
  3. Back Matter
    Pages 135-164

About this book

Introduction

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.

Keywords

Combinatorics cls commutative algebra matroid topology

Authors and affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridge

Bibliographic information

  • Book Title Combinatorics and Commutative Algebra
  • Authors Richard P. Stanley
  • Series Title Progress in Mathematics
  • Series Abbreviated Title Progress in Mathematics(Birkhäuser)
  • DOI https://doi.org/10.1007/b139094
  • Copyright Information Birkhäuser Boston 1996
  • Publisher Name Birkhäuser Boston
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-0-8176-3836-8
  • Softcover ISBN 978-0-8176-4369-0
  • eBook ISBN 978-0-8176-4433-8
  • Series ISSN 0743-1643
  • Series E-ISSN 2296-505X
  • Edition Number 2
  • Number of Pages IX, 166
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Combinatorics
    Commutative Rings and Algebras
    Topology
  • Buy this book on publisher's site