LEADER 02824nam a2200373 i 4500 001 991003265999707536 008 160801t2015 sz b 001 0 eng d 020 $a9783319194936 035 $ab14305811-39ule_inst 040 $aBibl. Dip.le Aggr. Matematica e Fisica - Sez. Matematica$beng 082 04$a512.4$223 084 $aAMS 13-02 084 $aAMS 03F65 084 $aAMS 13C10 084 $aAMS 13D02 084 $aAMS 13P10 084 $aLC QA251.3.Y46 100 1 $aYengui, Ihsen$0718166 245 10$aConstructive commutative algebra :$bprojective modules over polynomial rings and dynamical Gröbner bases /$cIhsen Yengui 260 $aCham [Switzerland] :$bSpringer,$cc2014 300 $avii, 271 p. ;$c24 cm 440 0$aLecture notes in mathematics,$x0075-8434 ;$v2138 504 $aIncludes bibliographical references (pages 259-268) and index 505 0 $aProjective modules over polynomial rings ; Dynamical Gröbner bases ; Syzygies in polynomial rings over valuation domains ; Exercises ; Detailed solutions to the exercises 520 $aThe main goal of this book is to find the constructive content hidden in abstract proofs of concrete theorems in Commutative Algebra, especially in well-known theorems concerning projective modules over polynomial rings (mainly the Quillen-Suslin theorem) and syzygies of multivariate polynomials with coefficients in a valuation ring. Simple and constructive proofs of some results in the theory of projective modules over polynomial rings are also given, and light is cast upon recent progress on the Hermite ring and Gröbner ring conjectures. New conjectures on unimodular completion arising from our constructive approach to the unimodular completion problem are presented. Constructive algebra can be understood as a first preprocessing step for computer algebra that leads to the discovery of general algorithms, even if they are sometimes not efficient. From a logical point of view, the dynamical evaluation gives a constructive substitute for two highly nonconstructive tools of abstract algebra: the Law of Excluded Middle and Zorn's Lemma. For instance, these tools are required in order to construct the complete prime factorization of an ideal in a Dedekind ring, whereas the dynamical method reveals the computational content of this construction. These lecture notes follow this dynamical philosophy 650 0$aCommutative algebra 650 0$aGröbner bases 650 0$aPolynomial rings 907 $a.b14305811$b22-11-16$c01-08-16 912 $a991003265999707536 945 $aLE013 13-XX YEN11 (2015)$g1$i2013000293912$lle013$op$pE46.79$q-$rl$s- $t0$u1$v0$w1$x0$y.i15789135$z22-11-16 996 $aConstructive commutative algebra$91392311 997 $aUNISALENTO 998 $ale013$b01-08-16$cm$da $e-$feng$gsz $h0$i0