Overview
- Presents a new point of view concerning problems in Commutative Algebra and Algebraic Geometry
- All the proofs are constructive (algorithms) and simple
- The reader can easily implement the presented algorithms in his preferred Computer Algebra System
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2138)
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Table of contents (6 chapters)
Keywords
About this book
The main goal of this book is to find the constructive content hidden in abstract proofs of concrete theorems in Commutative Algebra, especially in well-known theorems concerning projective modules over polynomial rings (mainly the Quillen-Suslin theorem) and syzygies of multivariate polynomials with coefficients in a valuation ring.
Simple and constructive proofs of some results in the theory of projective modules over polynomial rings are also given, and light is cast upon recent progress on the Hermite ring and Gröbner ring conjectures. New conjectures on unimodular completion arising from our constructive approach to the unimodular completion problem are presented.
Constructive algebra can be understood as a first preprocessing step for computer algebra that leads to the discovery of general algorithms, even if they are sometimes not efficient. From a logical point of view, the dynamical evaluation gives a constructive substitute for two highly nonconstructive toolsof abstract algebra: the Law of Excluded Middle and Zorn's Lemma. For instance, these tools are required in order to construct the complete prime factorization of an ideal in a Dedekind ring, whereas the dynamical method reveals the computational content of this construction. These lecture notes follow this dynamical philosophy.
Reviews
“Each chapter is self-contained. All the proofs are constructive and many illustrative examples are included. The book is well-written and easy to read. The book also contains some original and significant research results of the author. The book is highly recommended for researchers and students interesting in constructive commutative algebra.” (Chen Sheng, zbMATH 1360.13002, 2017)
“The purpose of this book is twofold: to find the computational content hidden in abstract proofs of concrete theorems in commutative algebra, and to give general algorithms for solving theorems of abstract algebra. … The book is based on lectures on constructive algebra that the author previously gave on two different occasions. Each chapter is self-contained. All the proofs are constructive and many illustrative examples are included.” (HaohaoWang, Mathematical Reviews, August, 2016)
Authors and Affiliations
Bibliographic Information
Book Title: Constructive Commutative Algebra
Book Subtitle: Projective Modules Over Polynomial Rings and Dynamical Gröbner Bases
Authors: Ihsen Yengui
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-3-319-19494-3
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing Switzerland 2015
Softcover ISBN: 978-3-319-19493-6Published: 12 December 2015
eBook ISBN: 978-3-319-19494-3Published: 11 December 2015
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: VII, 271
Number of Illustrations: 5 b/w illustrations
Topics: Commutative Rings and Algebras, Mathematical Logic and Foundations, Symbolic and Algebraic Manipulation