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Binomial Ideals

✍ Scribed by Jurgen Herzog, Takayuki Hibi, Hidefumi Ohsugi

Publisher
Springer
Year
2018
Tongue
English
Leaves
332
Series
Graduate Texts in Mathematics
Category
Library
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✦ Synopsis

This textbook provides an introduction to the combinatorial and statistical aspects of commutative algebra with an emphasis on binomial ideals. In addition to thorough coverage of the basic concepts and theory, it explores current trends, results, and applications of binomial ideals to other areas of mathematics.
The book begins with a brief, self-contained overview of the modern theory of Gröbner bases and the necessary algebraic and homological concepts from commutative algebra. Binomials and binomial ideals are then considered in detail, along with a short introduction to convex polytopes. Chapters in the remainder of the text can be read independently and explore specific aspects of the theory of binomial ideals, including edge rings and edge polytopes, join-meet ideals of finite lattices, binomial edge ideals, ideals generated by 2-minors, and binomial ideals arising from statistics. Each chapter concludes with a set of exercises and a list of related topics and results that will complement and offer a better understanding of the material presented.
Binomial Ideals is suitable for graduate students in courses on commutative algebra, algebraic combinatorics, and statistics. Additionally, researchers interested in any of these areas but familiar with only the basic facts of commutative algebra will find it to be a valuable resource.

✦ Table of Contents

Front Matter ....Pages i-xix
Front Matter ....Pages 1-1
Polynomial Rings and Gröbner Bases (Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi)....Pages 3-34
Review of Commutative Algebra (Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi)....Pages 35-58
Front Matter ....Pages 59-59
Introduction to Binomial Ideals (Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi)....Pages 61-86
Convex Polytopes and Unimodular Triangulations (Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi)....Pages 87-114
Front Matter ....Pages 115-115
Edge Polytopes and Edge Rings (Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi)....Pages 117-140
Join-Meet Ideals of Finite Lattices (Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi)....Pages 141-170
Binomial Edge Ideals and Related Ideals (Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi)....Pages 171-238
Ideals Generated by 2-Minors (Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi)....Pages 239-270
Statistics (Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi)....Pages 271-305
Back Matter ....Pages 307-321

✦ Subjects

Algebra, Commutative Algebra, Grobner Bases

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