Constructive commutative algebra : projective modules over polynomial rings and dynamical Gröbner bases / Ihsen Yengui.Material type: TextSeries: Lecture notes in mathematics (Springer-Verlag): 2138.Publisher: Cham : Springer, Copyright date: ©2015Description: 1 online resource (vii, 271 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 3319194941; 9783319194943Subject(s): Polynomial rings | Gröbner bases | Commutative algebra | Mathematics | Commutative Rings and Algebras | Mathematical Logic and Foundations | Symbolic and Algebraic Manipulation | Commutative algebra | Gröbner bases | Polynomial ringsGenre/Form: Electronic books. Additional physical formats: Printed edition:: No titleDDC classification: 512/.4 LOC classification: QA251.3 | .Y46 2015ebOnline resources: Click here to access online
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Includes bibliographical references (pages 259-268) and index.
Online resource; title from PDF title page (SpringerLink, viewed Dec. 16, 2015).
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.
Projective modules over polynomial rings -- Dynamical Gr¨obner bases -- Syzygies in polynomial rings over valuation domains -- Exercises -- Detailed solutions to the exercises.