Combinatorial Commutative Algebra (Graduate Texts in Mathematics, 227)

Recent developments are covered Contains over 100 figures and 250 exercises Includes complete proofs

Science & Math

Sturmfels, Bernd

Miller, Ezra

Mathematics

Springer

Oriented matroids are a very natural mathematical concept which presents itself in many different guises, and which has connections and applications to many different areas. These include discrete and computational geometry, combinatorics, convexity, topology, algebraic geometry, operations research, computer science and theoretical chemistry. This is the first comprehensive and accessible account of the subject. This book is intended for a diverse audience: graduate students who wish to learn the subject from scratch, researchers in the various fields of application who want to concentrate on certain aspects of the theory, specialists who need a thorough reference work, and others at points in between. A list of exercises and open problems ends each chapter, and the work is rounded off by an up-to-date and exhaustive reference list.

Oriented Matroids (Encyclopedia of Mathematics and its Applications, Series Number 46)

Reference

Björner, Anders

Las Vergnas, Michel

Ziegler, G|nter M.

White,Neil

Encyclopedias & Subject Guides

Cambridge University Press

This second edition of the first comprehensive, accessible account of the subject is intended for a diverse audience: graduate students who wish to learn the subject, researchers in the various fields of application who want to concentrate on certain theoretical aspects, and specialists who need a thorough reference work. For the second edition, the authors have greatly expanded the bibliography to ensure that it is comprehensive and up-to-date, and have also added an appendix surveying research since the first edition. A list of exercises and open problems ends each chapter.

Ziegler, Günter M.

Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of information about their classical counterparts. Tropical geometry is a young subject that has undergone a rapid development since the beginning of the 21st century. While establishing itself as an area in its own right, deep connections have been made to many branches of pure and applied mathematics. This book offers a self-contained introduction to tropical geometry, suitable as a course text for beginning graduate students. Proofs are provided for the main results, such as the Fundamental Theorem and the Structure Theorem. Numerous examples and explicit computations illustrate the main concepts. Each of the six chapters concludes with problems that will help the readers to practice their tropical skills, and to gain access to the research literature.

Introduction to Tropical Geometry (Graduate Studies in Mathematics, 161)

Maclagan, Diane

Amer Mathematical Society

Invitation to Nonlinear Algebra (Graduate Studies in Mathematics, 211)

Nonlinear algebra provides modern mathematical tools to address challenges arising in the sciences and engineering. It is useful everywhere, where polynomials appear: in particular, data and computational sciences, statistics, physics, optimization. The book offers an invitation to this broad and fast-developing area. It is not an extensive encyclopedia of known results, but rather a first introduction to the subject, allowing the reader to enter into more advanced topics. It was designed as the next step after linear algebra and well before abstract algebraic geometry. The book presents both classical topics--like the Nullstellensatz and primary decomposition--and more modern ones--like tropical geometry and semidefinite programming. The focus lies on interactions and applications. Each of the thirteen chapters introduces fundamental concepts. The book may be used for a one-semester course, and the over 200 exercises will help the readers to deepen their understanding of the subject.

Michalek, Mateusz

Metric Algebraic Geometry (Oberwolfach Seminars)

Computers & Technology

Breiding, Paul

Kohn, Kathlén

Computer Science

Birkhauser

This book is both an easy-to-read textbook for invariant theory and a challenging research monograph that introduces a new approach to the algorithmic side of invariant theory. Students will find the book an easy introduction to this "classical and new" area of mathematics. Researchers in mathematics, symbolic computation, and computer science will get access to research ideas, hints for applications, outlines and details of algorithms, examples and problems.

Algorithms in Invariant Theory (Texts & Monographs in Symbolic Computation)

Springer-Verlag Gmbh

Combinatorics and Algebraic Geometry have enjoyed a fruitful interplay since the nineteenth century. Classical interactions include invariant theory, theta functions and enumerative geometry. The aim of this volume is to introduce recent developments in combinatorial algebraic geometry and to approach algebraic geometry with a view towards applications, such as tensor calculus and algebraic statistics. A common theme is the study of algebraic varieties endowed with a rich combinatorial structure. Relevant techniques include polyhedral geometry, free resolutions, multilinear algebra, projective duality and compactifications.

Combinatorial Algebraic Geometry: Levico Terme, Italy 2013, Editors: Sandra Di Rocco, Bernd Sturmfels (C.I.M.E. Foundation Subseries)

Conca, Aldo

Di Rocco, Sandra

Draisma, Jan

Huh, June

Viviani, Filippo

The theory of Gröbner bases is a main tool for dealing with rings of differential operators. This book reexamines the concept of Gröbner bases from the point of view of geometric deformations. The algorithmic methods introduced in this book are particularly useful for studying the systems of multidimensional hypergeometric PDE's introduced by Gelfand, Kapranov, and Zelevinsky. A number of original research results are contained in the book, and many open problems are raised for future research in this rapidly growing area of computational mathematics.

Groebner Deformations of Hypergeometric Differential Equations, Algorithms and Computation in Mathematics, Volume 6 (Algorithms and Computation in Mathematics, 6)

Saito, Mutsumi

Takayama, Nobuki

Programming

The theory of Gröbner bases is a main tool for dealing with rings of differential operators. This book reexamines the concept of Gröbner bases from the point of view of geometric deformations. The algorithmic methods introduced in this book are particularly useful for studying the systems of multidimensional hypergeometric Pde's introduced by Gelfand, Kapranov, and Zelevinsky. A number of original research results are contained in the book, and many open problems are raised for future research in this rapidly growing area of computational mathematics.

Gröbner Deformations of Hypergeometric Differential Equations (Algorithms and Computation in Mathematics)

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