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Combinatorics and Commutative Algebra [Paperback]

Richard Stanley
2.5 out of 5 stars  See all reviews (2 customer reviews)
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Book Description

Oct. 15 2004 0817643699 978-0817643690 2nd ed. 1996

* Stanley represents a broad perspective with respect to two significant topics from Combinatorial Commutative Algebra:

1) The theory of invariants of a torus acting linearly on a polynomial ring, and

2) The face ring of a simplicial complex

* In this new edition, the author further develops some interesting properties of face rings with application to combinatorics

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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.

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The purpose of this introduction is to provide the reader with the relevant background from combinatorics, algebra, and topology for understanding of the text. Read the first page
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Most helpful customer reviews
4.0 out of 5 stars Green book Jan. 9 2002
Stanley's "green book" is an overview of a new aspect in commutative algebra:its links with combinatorics.Stanley,as one of the pionners on the subject,is the right one to give us this compendius.But,unfortunally,its hard to beguiners.
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1.0 out of 5 stars I dont read this book yet! March 2 2001
I dont read this book yet!
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Amazon.com: 4.0 out of 5 stars  2 reviews
4 of 4 people found the following review helpful
4.0 out of 5 stars great book Feb. 12 2006
By N. Davison - Published on Amazon.com
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 - Published on Amazon.com
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.
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