Combinatorics and Commutative Algebra (Progress in Mathematics) [Hardcover]

Richard P. Stanley (Author)

Book Description

ISBN-10: 0817638369 | ISBN-13: 978-0817638368 | Publication Date: March 1, 1996 | Edition: 2nd rev. and enlarged ed.

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.

Editorial Reviews

From the Back Cover

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. --This text refers to the Paperback edition.

Product Details

• Hardcover: 168 pages
• Publisher: Birkhäuser Boston; 2nd rev. and enlarged ed. edition (March 1, 1996)
• Language: English
• ISBN-10: 0817638369
• ISBN-13: 978-0817638368
• Product Dimensions: 9 x 6 x 1 inches
• Shipping Weight: 1.4 pounds
• Average Customer Review: 4.0 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #4,211,287 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

4 of 4 people found the following review helpful:

4.0 out of 5 stars great book, February 12, 2006

By 

N. Davison - See all my reviews
(REAL NAME)   

This review is from: Combinatorics and Commutative Algebra (Progress in Mathematics) (Paperback)

Great book - just be ready to get hold of some other books to fill in the details. If you're not entirely comfortable with the stuff in chapter 0 concerning modules (projectivity, injectivity), tor and ext, and homology from topology, maybe find a quick introduction to these elsewhere - while its certainly possible to learn about these things from chapter 0 of this book, it's not exactly the most painless way to do it. It was my first book on the subject, then I got Miller and Sturmfels: it would make more sense to reverse that order. Better still, get both simultaneously, and when the Stanley becomes a bit dense refer to M+S. I really loved this book, but it was damn hard at times!

4 of 4 people found the following review helpful:

4.0 out of 5 stars This is a research monograph!, April 19, 2005

By 

Loopspace (Norway) - See all my reviews

This review is from: Combinatorics and Commutative Algebra (Progress in Mathematics) (Paperback)

For those looking for a thorough introduction to the theory in this book, I would suggest a look at Miller and Sturmfels recent book on combinatorial commutative algebra or the book on Cohen-Macaulay rings by Bruns and Herzog. This is a great book, but I don't think it was intended as a beginners first book . Instead its style concentrates on presenting results found in research-articles ( prior to this book, there doesn't seem to have been any texts giving an overview of the theory ). It is great for looking up results, reading the needed prerequisites without going into details and finding important references.