Synopsis
* 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
"synopsis" may belong to another edition of this title.
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.
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Combinatorics and Commutative Algebra (Progress in Mathematics). Second edition
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Combinatorics and Commutative Algebra. (Progress in mathematics, vol.41)
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Boston. Birkhäuser., 1996
ISBN 10: 0817638369
ISBN 13: 9780817638368
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