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020 _a9783319194943
_9978-3-319-19494-3
024 7 _a10.1007/978-3-319-19494-3
_2doi
050 4 _aQA251.3
072 7 _aPBF
_2bicssc
072 7 _aMAT002010
_2bisacsh
072 7 _aPBF
_2thema
082 0 4 _a512.44
_223
100 1 _aYengui, Ihsen.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
245 1 0 _aConstructive Commutative Algebra
_h[electronic resource] :
_bProjective Modules Over Polynomial Rings and Dynamical Gröbner Bases /
_cby Ihsen Yengui.
250 _a1st ed. 2015.
264 1 _aCham :
_bSpringer International Publishing :
_bImprint: Springer,
_c2015.
300 _aVII, 271 p. 5 illus.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aLecture Notes in Mathematics,
_x0075-8434 ;
_v2138
505 0 _aProjective modules over polynomial rings -- Dynamical Gr ̈obner 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 _aCommutative rings.
650 0 _aMathematical logic.
650 0 _aComputer science—Mathematics.
650 1 4 _aCommutative Rings and Algebras.
_0https://scigraph.springernature.com/ontologies/product-market-codes/M11043
650 2 4 _aMathematical Logic and Foundations.
_0https://scigraph.springernature.com/ontologies/product-market-codes/M24005
650 2 4 _aSymbolic and Algebraic Manipulation.
_0https://scigraph.springernature.com/ontologies/product-market-codes/I17052
710 2 _aSpringerLink (Online service)
773 0 _tSpringer Nature eBook
776 0 8 _iPrinted edition:
_z9783319194950
776 0 8 _iPrinted edition:
_z9783319194936
830 0 _aLecture Notes in Mathematics,
_x0075-8434 ;
_v2138
856 4 0 _uhttps://doi.org/10.1007/978-3-319-19494-3
912 _aZDB-2-SMA
912 _aZDB-2-SXMS
912 _aZDB-2-LNM
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