This book provides a quick access to computational tools for algebraic geometry, the mathematical discipline which handles solution sets of polynomial equations. Originating from a number of intense one week schools taught by the authors, the text is designed so as to provide a step by step introduction which enables the reader to get started with his own computational experiments right away. The authors present the basic concepts and ideas in a compact way.

Singularity theory is a young, rapidly-growing topic with connections to algebraic geometry, complex analysis, commutative algebra, representations theory, Lie groups theory and topology, and many applications in the natural and technical sciences. This book presents the basic singularity theory of analytic spaces, including local deformation theory and the theory of plane curve singularities. It includes complete proofs.

From the reviews of the first edition:

"It is certainly no exaggeration to say that A Singular Introduction to Commutative Algebra aims to lead a further stage in the computational revolution in commutative algebra . Among the great strengths and most distinctive features is a new, completely unified treatment of the global and local theories. making it one of the most flexible and most efficient systems of its type....another strength of Greuel and Pfister's book is its breadth of coverage of theoretical topics in the portions of commutative algebra closest to algebraic geometry, with algorithmic treatments of almost every topic....Greuel and Pfister have written a distinctive and highly useful book that should be in the library of every commutative algebraist and algebraic geometer, expert and novice alike."

J.B. Little, MAA, March 2004

The second edition is substantially enlarged by a chapter on Groebner bases in non-commtative rings, a chapter on characteristic and triangular sets with applications to primary decomposition and polynomial solving and an appendix on polynomial factorization including factorization over algebraic field extensions and absolute factorization, in the uni- and multivariate case."

Singular algebraic curves have been in the focus of study in algebraic geometry from the very beginning, and till now remain a subject of an active research related to many modern developments in algebraic geometry, symplectic geometry, and tropical geometry. The monograph suggests a unified approach to the geometry of singular algebraic curves on algebraic surfaces and their families, which applies to arbitrary singularities, allows one to treat all main questions concerning the geometry of equisingular families of curves, and, finally, leads to results which can be viewed as the best possible in a reasonable sense. Various methods of the cohomology vanishing theory as well as the patchworking construction with its modifications will be of a special interest for experts in algebraic geometry and singularity theory. The introductory chapters on zero-dimensional schemes and global deformation theory can well serve as a material for special courses and seminars for graduate and post-graduate students.Geometry in general plays a leading role in modern mathematics, and algebraic geometry is the most advanced area of research in geometry. In turn, algebraic curves for more than one century have been the central subject of algebraic geometry both in fundamental theoretic questions and in applications to other fields of mathematics and mathematical physics. Particularly, the local and global study of singular algebraic curves involves a variety of methods and deep ideas from geometry, analysis, algebra, combinatorics and suggests a number of hard classical and newly appeared problems which inspire further development in this research area.

This book provides a quick access to computational tools for algebraic geometry, the mathematical discipline which handles solution sets of polynomial equations. Originating from a number of intense one week schools taught by the authors, the text is designed so as to provide a step by step introduction which enables the reader to get started with his own computational experiments right away. The authors present the basic concepts and ideas in a compact way.

Singularity theory is a young, rapidly-growing topic with connections to algebraic geometry, complex analysis, commutative algebra, representations theory, Lie groups theory and topology, and many applications in the natural and technical sciences. This book presents the basic singularity theory of analytic spaces, including local deformation theory and the theory of plane curve singularities. It includes complete proofs.

This substantially enlarged second edition aims to lead a further stage in the computational revolution in commutative algebra. This is the first handbook/tutorial to extensively deal with SINGULAR. Among the book's most distinctive features is a new, completely unified treatment of the global and local theories. Another feature of the book is its breadth of coverage of theoretical topics in the portions of commutative algebra closest to algebraic geometry, with algorithmic treatments of almost every topic.

This volume surveys important topics in singularity theory, with a particular focus on computational aspects of the subject. The contributors to this volume include R. O. Buchweitz, Y. A. Drozd, W. Ebeling, H. A. Hamm, Le D. T., I. Luengo, F.-O. Schreyer, E. Shustin, J. H. M. Steenbrink, D. van Straten, B. Teissier and J. Wahl. Together they describe the development of various areas of singularity theory over many years, and a range of open questions are discussed. Research workers in singularity theory, computer algebra or related subjects will find that this book contains a wealth of valuable information.