02824nam a2200373 i 4500991003265999707536160801t2015 sz b 001 0 eng d9783319194936b14305811-39ule_instBibl. Dip.le Aggr. Matematica e Fisica - Sez. Matematicaeng512.423AMS 13-02AMS 03F65AMS 13C10AMS 13D02AMS 13P10LC QA251.3.Y46Yengui, Ihsen718166Constructive commutative algebra :projective modules over polynomial rings and dynamical Gröbner bases /Ihsen YenguiCham [Switzerland] :Springer,c2014vii, 271 p. ;24 cmLecture notes in mathematics,0075-8434 ;2138Includes bibliographical references (pages 259-268) and indexProjective modules over polynomial rings ; Dynamical Gröbner bases ; Syzygies in polynomial rings over valuation domains ; Exercises ; Detailed solutions to the exercisesThe 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 philosophyCommutative algebraGröbner basesPolynomial rings.b1430581122-11-1601-08-16991003265999707536LE013 13-XX YEN11 (2015)12013000293912le013pE46.79-l- 01010.i1578913522-11-16Constructive commutative algebra1392311UNISALENTOle01301-08-16ma -engsz 00