Monomial Ideals

  • Jürgen Herzog
  • Takayuki Hibi

Table of contents

  1. Front Matter
    Pages I-XVI
  2. Gröbner bases

    1. Front Matter
      Pages 1-1
    2. Jürgen Herzog, Takayuki Hibi
      Pages 3-22
    3. Jürgen Herzog, Takayuki Hibi
      Pages 23-40
    4. Jürgen Herzog, Takayuki Hibi
      Pages 41-50
    5. Jürgen Herzog, Takayuki Hibi
      Pages 51-74
    6. Jürgen Herzog, Takayuki Hibi
      Pages 75-93
  3. Hilbert functions and resolutions

    1. Front Matter
      Pages 95-95
    2. Jürgen Herzog, Takayuki Hibi
      Pages 97-113
    3. Jürgen Herzog, Takayuki Hibi
      Pages 115-128
    4. Jürgen Herzog, Takayuki Hibi
      Pages 129-149
  4. Combinatorics

    1. Front Matter
      Pages 151-151
    2. Jürgen Herzog, Takayuki Hibi
      Pages 153-182
    3. Jürgen Herzog, Takayuki Hibi
      Pages 183-210
    4. Jürgen Herzog, Takayuki Hibi
      Pages 211-236
    5. Jürgen Herzog, Takayuki Hibi
      Pages 237-261
  5. Back Matter
    Pages 263-305

About this book


This book demonstrates current trends in research on combinatorial and computational commutative algebra with a primary emphasis on topics related to monomial ideals.

Providing a useful and quick introduction to areas of research spanning these fields, Monomial Ideals is split into three parts. Part I offers a quick introduction to the modern theory of Gröbner bases as well as the detailed study of generic initial ideals. Part II supplies Hilbert functions and resolutions and some of the combinatorics related to monomial ideals including the Kruskal—Katona theorem and algebraic aspects of Alexander duality. Part III discusses combinatorial applications of monomial ideals, providing a valuable overview of some of the central trends in algebraic combinatorics.

Main subjects include edge ideals of finite graphs, powers of ideals, algebraic shifting theory and an introduction to discrete polymatroids. Theory is complemented by a number of examples and exercises throughout, bringing the reader to a deeper understanding of concepts explored within the text.

Self-contained and concise, this book will appeal to a wide range of readers, including PhD students on advanced courses, experienced researchers, and combinatorialists and non-specialists with a basic knowledge of commutative algebra.

Since their first meeting in 1985, Juergen Herzog (Universität Duisburg-Essen, Germany) and Takayuki Hibi (Osaka University, Japan), have worked together on a number of research projects, of which recent results are presented in this monograph.


Alexander duality Gröbner basis Hilbert function algebraic shifting generic initial ideal graded Betti number minimal free resolution simplicial complex

Authors and affiliations

  • Jürgen Herzog
    • 1
  • Takayuki Hibi
    • 2
  1. 1.Fachbereich MathematikUniversität Duisburg-Essen Fachbereich MathematikEssenGermany
  2. 2.Department of Pure & Applied MathematicsOsaka University Graduate School of Inf. Science & Techn.ToyonakaJapan

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag London Limited 2011
  • Publisher Name Springer, London
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-0-85729-105-9
  • Online ISBN 978-0-85729-106-6
  • About this book