Monomial Ideals and Their Decompositions

W. Frank Moore, Mark Rogers, Sean Sather-Wagstaff (Autoren)

Buch | Softcover

XXIV, 387 Seiten

2018 | 1st ed. 2018
Springer International Publishing (Verlag)
978-3-319-96874-2 (ISBN)

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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.

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.

-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.

"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)

Erscheinungsdatum	04.11.2018
Reihe/Serie	Universitext
Zusatzinfo	XXIV, 387 p. 55 illus.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	629 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Schlagworte	algorithms • Cohen-Macaulayness • combinatorial commutative algebra • Commutative algebra • decompositions of monomial ideals • Dickson's Lemma • Dickson’s Lemma • edge ideal construction of Villarreal • Hilbert functions • irreducible decompositions • Macaulay 2 • m-irreducible decompositions • modifying monomial ideals • monomial ideals • parametric decompositions • Phasor Measurement Units • polynomial rings • Simplicial Complexes • Stanley-Reisner ideals • vertex covers
ISBN-10	3-319-96874-2 / 3319968742
ISBN-13	978-3-319-96874-2 / 9783319968742
Zustand	Neuware