Monomial Ideals and Their Decompositions

ISBN-13: 9783319968742 / Angielski / Miękka / 2018 / 387 str.

William Francis Moore; Mark Wayne Rogers; Sean Michael Sather-Wagstaff

Opis

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.

Recenzje

"The present book is thought as a gentle introduction to monomial ideals ... . All the chapters contain exercises and Macaulay 2 material for the computational exploration of the presented notions." (Christos Tatakis, zbMATH 1476.13002, 2022)

"Primarily directed at advanced undergraduates, the text is also a valuable resource for graduate students and researchers who wish to learn more about the subject, providing an introduction to active research topics in combinatorial commutative algebra and its applications. ... the authors' presentation of monomial decompositions and their applications is an exciting, enlightening read and will serve an individual reader or class instructor well." (Timothy B. P. Clark, Mathematical Reviews, October, 2019)
"Each definition includes examples of reasonably common structures ... . This style makes the text accessible to advanced undergraduates. ... it will be useful to those who work in symbolic computation and theory." (Paul Cull, Computing Reviews, May 13, 2019)

Spis treści

-Introduction.- 1. Fundamental Properties of Monomial Ideals . -2. Operations on Monomial Ideals.- 3.  M-Irreducible Ideals and Decompositions.- 4.  Connections with Combinatorics.- 5.  Connections with Other Areas. -6.  Parametric Decompositions of Monomial Ideals.- 7.  Computing M-Irreducible Decompositions.- Appendix A. Foundational Concepts.- Appendix B. Introduction to Macaulay2.- Bibliography.- Index.

 

O autorze

W. Frank Moore is an Associate Professor of Mathematics at Wake Forest University. He earned his PhD from the University of Nebraska-Lincoln, and his research is in the homological algebra of commutative and noncommutative rings.

Mark Rogers is a Professor in the Department of Mathematics at Missouri State University. He earned his PhD from Purdue University, and his area of research is commutative algebra.

Sean Sather-Wagstaff is an Associate Professor in Clemson University’s department of Mathematical Sciences. He earned his PhD from the University of Utah, specializing in homological commutative algebra.

Blurb

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.

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