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Binomial Ideals

  • Jürgen Herzog
  • Takayuki Hibi
  • Hidefumi Ohsugi

Part of the Graduate Texts in Mathematics book series (GTM, volume 279)

Table of contents

  1. Front Matter
    Pages i-xix
  2. Basic Concepts

    1. Front Matter
      Pages 1-1
    2. Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi
      Pages 3-34
    3. Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi
      Pages 35-58
  3. Binomial Ideals and Convex Polytopes

    1. Front Matter
      Pages 59-59
    2. Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi
      Pages 61-86
    3. Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi
      Pages 87-114
  4. Applications in Combinatorics and Statistics

    1. Front Matter
      Pages 115-115
    2. Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi
      Pages 117-140
    3. Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi
      Pages 141-170
    4. Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi
      Pages 171-238
    5. Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi
      Pages 239-270
    6. Jürgen Herzog, Takayuki Hibi, Hidefumi Ohsugi
      Pages 271-305
  5. Back Matter
    Pages 307-321

About this book

Introduction

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.

Keywords

binomial ideals commutative algebra Gröbner bases convex polytopes toricideals join-meet ideals algebraic statistics

Authors and affiliations

  • Jürgen Herzog
    • 1
  • Takayuki Hibi
    • 2
  • Hidefumi Ohsugi
    • 3
  1. 1.Fakultät für MathematikUniversität Duisburg-EssenEssenGermany
  2. 2.Department of Pure & Applied MathematicsGraduate School of Information Science and Technology, Osaka UniversitySuitaJapan
  3. 3.Department of Mathematical SciencesSchool of Science and Technology, Kwansei Gakuin UniversitySandaJapan

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-95349-6
  • Copyright Information Springer International Publishing AG, part of Springer Nature 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-95347-2
  • Online ISBN 978-3-319-95349-6
  • Series Print ISSN 0072-5285
  • Series Online ISSN 2197-5612
  • Buy this book on publisher's site