Binomial Ideals

Beschreibung

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 witha 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.

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.
Presents a thorough study of binomial ideals and their applications, working from the basics through to current research Offers an accessible introduction to the area for combinatorialists and statisticians, building only on the basics of commutative algebra. Explores the new research area of algebraic statistics and its relation to toric ideals and their Gröbner bases

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Jürgen Herzog

Themen in »Binomial Ideals«

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

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“This is a valuable resource for students and researchers entering this area of combinatorial commutative algebra.” (Thomas Kahle, Mathematical Reviews, November, 2019)
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