Constructive Commutative Algebra
Projective Modules Over Polynomial Rings and Dynamical Gröbner Bases
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Beschreibung
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.
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.
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
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Ihsen Yengui
Themen in »Constructive Commutative Algebra«
13Cxx,13Pxx,14Qxx,03Fxx. Dynamical Gröbner Bases Dynamical Methods in Commutative Algebra Gröbner Ring Conjecture Hermite Ring Conjecture Quillen-Suslin Theorem
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“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)
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