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Computing in algebraic geometry. A quick start using SINGULAR. (English) Zbl 1095.14001
Algorithms and Computation in Mathematics 16. Berlin: Springer; New Delhi: Hindustan Book Agency (ISBN 3-540-28992-5/hbk). xvi, 327 p. (2006).
Classical algebraic geometry is closely related to the problem of solving polynomial equations. In fact a (projective) variety over an algebraically closed field is defined as the zero set of a family of (homogeneous) polynomials. Besides of the relations of algebraic geometry to the advanced technique of schemes and cohomology there is a growing number of research papers in algebraic geometry originated from explicit computations. A technical tool for attacking this kind of concrete problems is developed by the theory of Gröbner bases and their implementation in several computer algebra systems, among them Singular and Macaulay.
The book under review may serve as a quick start to computing in algebraic geometry. There is a presentation of the background material illustrated by several examples of concrete computations. To this end the authors simultaneously introduce the computer algebra system Singular and which may serve as a tool for samples for computations carried out by an interested reader. The text may accompany students of beginner’s courses in algebraic geometry, by master and PhD students experimenting with examples related to their interests, as well as more experienced researchers in order to provide powerful computational methods in their discipline.
The material of the book grows out of several courses given by the authors. The text is organized as an introductory lecture, nine lectures, five practical sessions and an appendix. In the introductory lecture the authors present remarks on the development of computer algebra. The first two lectures are devoted historical remarks to Gröbner bases and the computational problems arising from the geometry-algebra dictionary and their solutions with Gröbner bases. Beginning with samples in the first two lectures there is a thorough introduction to Singular in lecture 3. Lectures 4 and 5 treat computations in homological algebra. Lecture 6 is devoted to methods for exact and symbolic-numerical solutions of polynomial equations.
The last two lectures are entitled “Algorithms for Invariant Theory” and “Computing in Local Rings”. The material of the lectures is completed by five problem sessions with exercises and some hints to their solutions in the appendix. This enables a beginner the more tricky use of Singular ending with an introduction how to write libraries for Singular. In the appendix the authors include an additional lecture on computing sheaf cohomology and Beilinson monads. This part of the appendix in relation with the computational cohomology of the lectures 4 and 5 is the highlight of this book. It provides the bridge of solving polynomial equations, the traditional part of algebraic geometry, to the advanced technique of sheaf cohomology in algebraic geometry and its computation. The prerequisites for the readers are basic knowledge in algebraic geometry and related commutative algebra as one might found it in the textbook by [D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms, Undergraduate Texts in Mathematics. (1996; Zbl 0861.13012)]. In the course of the book the reader will become more familiar with advanced algebraic geometry and its computational aspects. For a Singular introduction to commutative algebra see also the book by [G.-M. Greuel and G. Pfister, A Singular introduction to commutative algebra. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann. (2002; Zbl 1023.13001)].

14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14Qxx Computational aspects in algebraic geometry
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
68W30 Symbolic computation and algebraic computation
Macaulay2; SINGULAR
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