Ideals Of Powers And Powers Of Ideals

This book discusses regular powers and symbolic powers of ideals from three perspectives– algebra, combinatorics and geometry – and examines the interactions between them. It invites readers to explore the evolution of the set of associated primes of higher and higher powers of an ideal and explains the evolution of ideals associated with combinatorial objects like graphs or hypergraphs in terms of the original combinatorial objects. It also addresses similar questions concerning our understanding of the Castelnuovo-Mumford regularity of powers of combinatorially defined ideals in terms of the associated combinatorial data. From a more geometric point of view, the book considers how the relations between symbolic and regular powers can be interpreted in geometrical terms. Other topics covered include aspects of Waring type problems, symbolic powers of an ideal and their invariants (e.g., the Waldschmidt constant, the resurgence), and the persistence of associated primes.

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.

Ideal for graduate students and researchers, this book presents a unified treatment of the central notions of integral closure.

Ten years of research uncover the secret source of growth and profit … Those who center their business on improving people’s lives have a growth rate triple that of competitors and outperform the market by a huge margin. They dominate their categories, create new categories and maximize profit in the long term. Pulling from a unique ten year growth study involving 50,000 brands, Jim Stengel shows how the world's 50 best businesses—as diverse as Method, Red Bull, Lindt, Petrobras, Samsung, Discovery Communications, Visa, Zappos, and Innocent—have a cause and effect relationship between financial performance and their ability to connect with fundamental human emotions, hopes, values and greater purposes. In fact, over the 2000s an investment in these companies—“The Stengel 50”—would have been 400 percent more profitable than an investment in the S&P 500. Grow is based on unprecedented empirical research, inspired (when Stengel was Global Marketing Officer of Procter & Gamble) by a study of companies growing faster than P&G. After leaving P&G in 2008, Stengel designed a new study, in collaboration with global research firm Millward Brown Optimor. This study tracked the connection over a ten year period between financial performance and customer engagement, loyalty and advocacy. Then, in a further investigation of what goes on in the “black box” of the consumer’s mind, Stengel and his team tapped into neuroscience research to look at customer engagement and measure subconscious attitudes to determine whether the top businesses in the Stengel Study were more associated with higher ideals than were others. Grow thus deftly blends timeless truths about human behavior and values into an action framework – how you discover, build, communicate, deliver and evaluate your ideal. Through colorful stories drawn from his fascinating personal experiences and “deep dives” that bring out the true reasons for such successes as the Pampers, HP, Discovery Channel, Jack Daniels and Zappos, Grow unlocks the code for twenty-first century business success.

This textbook on combinatorial commutative algebra focuses on properties of monomial ideals in polynomial rings and their connections with other areas of mathematics such as combinatorics, electrical engineering, topology, geometry, and homological algebra. Aimed toward advanced undergraduate students and graduate students who have taken a basic course in abstract algebra that includes polynomial rings and ideals, this book serves as a core text for a course in combinatorial commutative algebra or as preparation for more advanced courses in the area. The text contains over 600 exercises to provide readers with a hands-on experience working with the material; the exercises include computations of specific examples and proofs of general results. Readers will receive a firsthand introduction to the computer algebra system Macaulay2 with tutorials and exercises for most sections of the text, preparing them for significant computational work in the area. Connections to non-monomial areas of abstract algebra, electrical engineering, combinatorics and other areas of mathematics are provided which give the reader a sense of how these ideas reach into other areas.

The study of determinantal ideals and of classical determinantal rings is an old topic of commutative algebra. As in most of the cases, the theory evolved from algebraic geometry, and soon became an important topic in commutative algebra. Looking back, one can say that it is the merit of Eagon and Northcott to be the first who brought to the attention of algebraists the determinantal ideals and investigated them by the methods of commutative and homological algebra. Later on, Buchsbaum and Eisenbud, in a long series of articles, went further along the way of homological investigation of determinantal ideals, while Eagon and Hochster studied them using methods of commutative algebra in order to prove that the classical determinantal rings are normal Cohen-Macaulay domains. As shown later by C. DeConcini, D. Eisenbud, and C. Procesi the appropriate framework including all three types of rings is that of algebras with straightening law, and the standard monomial theory on which these algebras are based yields computationally effective results. A coherent treatment of determinantal ideals from this point of view was given by Bruns and Vetter in their seminal book. The author's book aims to a thorough treatment of all three types of determinantal rings in the light of the achievements of the last fifteen years since the publication of Bruns and Vetter's book. They implicitly assume that the reader is familiar with the basics of commutative algebra. However, the authors include some of the main notions and results from Bruns and Vetter's book for the sake of completeness, but without proofs. The authors recommend the reader to first look at the book of Bruns and Vetter in order to get a feel for the flavour of this field. The author's book is meant to be a reference text for the current state of research in the theory of determinantal rings. It was structured in such a way that it can be used as textbook for a one semester graduate course in advanced topics in Algebra, and at the PhD level.

Historically, the study of monomial ideals became fashionable after the pioneering work by Richard Stanley in 1975 on the upper bound conjecture for spheres. On the other hand, since the early 1990s, under the strong influence of Gröbner bases, binomial ideals became gradually fashionable in commutative algebra. The last ten years have seen a surge of research work in the study of monomial and binomial ideals. Remarkable developments in, for example, finite free resolutions, syzygies, Hilbert functions, toric rings, as well as cohomological invariants of ordinary powers, and symbolic powers of monomial and binomial ideals, have been brought forward. The theory of monomial and binomial ideals has many benefits from combinatorics and Göbner bases. Simultaneously, monomial and binomial ideals have created new and exciting aspects of combinatorics and Göbner bases. In the present Special Issue, particular attention was paid to monomial and binomial ideals arising from combinatorial objects including finite graphs, simplicial complexes, lattice polytopes, and finite partially ordered sets, because there is a rich and intimate relationship between algebraic properties and invariants of these classes of ideals and the combinatorial structures of their combinatorial counterparts. This volume gives a brief summary of recent achievements in this area of research. It will stimulate further research that encourages breakthroughs in the theory of monomial and binomial ideals. This volume provides graduate students with fundamental materials in this research area. Furthermore, it will help researchers find exciting activities and avenues for further exploration of monomial and binomial ideals. The editors express our thanks to the contributors to the Special Issue. Funds for APC (article processing charge) were partially supported by JSPS (Japan Society for the Promotion of Science) Grants-in-Aid for Scientific Research (S) entitled "The Birth of Modern Trends on Commutative Algebra and Convex Polytopes with Statistical and Computational Strategies" (JP 26220701). The publication of this volume is one of the main activities of the grant.