Commutative Algebra, Singularities and Computer Algebra presents current trends in commutative algebra, algebraic combinatorics, singularity theory and computer algebra, and highlights the interaction between these disciplines. Contributions by leading international mathematicians thoroughly discuss topics in: modules theory, integrally closed ideals and determinantal ideals, singularities in projective spaces and Castelnuovo-Mumford regularity, Groebner and SAGBI basis, and the use of the computer packages Bergman, CoCoA and SINGULAR.

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 Grobner 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 (Universitat 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."

Commutative Algebra, Singularities and Computer Algebra presents current trends in commutative algebra, algebraic combinatorics, singularity theory and computer algebra, and highlights the interaction between these disciplines. Contributions by leading international mathematicians thoroughly discuss topics in: modules theory, integrally closed ideals and determinantal ideals, singularities in projective spaces and Castelnuovo-Mumford regularity, Groebner and SAGBI basis, and the use of the computer packages Bergman, CoCoA and SINGULAR.

This book provides a concise yet comprehensive and self-contained introduction to Grobner basis theory and its applications to various current research topics in commutative algebra. It especially aims to help young researchers become acquainted with fundamental tools and techniques related to Grobner bases which are used in commutative algebra and to arouse their interest in exploring further topics such as toric rings, Koszul and Rees algebras, determinantal ideal theory, binomial edge ideals, and their applications to statistics. The book can be used for graduate courses and self-study. More than 100 problems will help the readers to better understand the main theoretical results and will inspire them to further investigate the topics studied in this book.

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 Groebner 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.

In the past two decades Cohen-Macaulay rings and modules have been central topics in commutative algebra. This book meets the need for a thorough, self-contained introduction to the subject. The authors emphasize the study of explicit, specific rings, making the presentation as concrete as possible. The general theory is applied to a number of examples and the connections with combinatorics are highlighted. Throughout each chapter, the authors have supplied many examples and exercises.