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Über dieses Buch

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 tools of 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.



Chapter 1. Introduction

Constructive algebra can be seen as an abstract version of computer algebra. In computer algebra, on the one hand, one attempts to construct efficient algorithms for solving concrete problems given in an algebraic formulation, where a problem is understood to be concrete if its hypotheses and conclusion have computational content. Constructive algebra, on the other hand, can be understood as a “preprocessing” step for computer algebra that leads to general algorithms, even if they are sometimes not efficient. In constructive algebra, one tries to give general algorithms for solving “virtually any” theorem of abstract algebra.

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Chapter 2. Projective Modules Over Polynomial Rings

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Chapter 3. Dynamical Gröbner Bases

The Euclidean division algorithm plays a key role when dealing with univariate polynomials with coefficients in a field K.

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Chapter 4. Syzygies in Polynomial Rings Over Valuation Domains

It is folklore (see for example Theorem 7.3.3 in [68]) that if V is a valuation domain, then V[X1, . . . ,X k ] (k ∈ ℕ) is coherent: that is, syzygy modules of finitely-generated ideals of V[X1, . . . ,X k ] are finitely-generated. The proof in the above-mentioned reference relies on a profound and difficult result published in a huge paper by Gruson and Raynaud [75]. There is nevertheless no known general algorithm for this remarkable result, and it seems difficult to compute the syzygy module even for small polynomials.

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Chapter 5. Exercises

Exercise 372. Prove constructively that a valuation domain has Krull dimension ≤ 1 if and only if its valuation group is archimedean.

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Chapter 6. Detailed Solutions to the Exercises

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