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020 _a9783031692130
_9978-3-031-69213-0
024 7 _a10.1007/978-3-031-69213-0
_2doi
050 4 _aQA564-609
072 7 _aPBMW
_2bicssc
072 7 _aMAT012010
_2bisacsh
072 7 _aPBMW
_2thema
100 1 _aScheiderer, Claus.
_eauthor.
_0(orcid)0000-0002-5452-1114
_1https://orcid.org/0000-0002-5452-1114
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_984157
245 1 2 _aA Course in Real Algebraic Geometry
_h[electronic resource] :
_bPositivity and Sums of Squares /
_cby Claus Scheiderer.
250 _a1st ed. 2024.
264 1 _aCham :
_bSpringer International Publishing :
_bImprint: Springer,
_c2024.
300 _aXVIII, 404 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 _aGraduate Texts in Mathematics,
_x2197-5612 ;
_v303
505 0 _a1 Ordered Fields -- 2 Positive Polynomials and Sums of Squares -- 3 The Real Spectrum -- 4 Semialgebraic Geometry -- 5 The Archimedean Property -- 6 Positive Polynomials with Zeros -- 7 Sums of Squares on Projective Varieties -- 8 Sums of Squares and Optimization -- Appendix A: Commutative Algebra and Algebraic Geometry -- Appendix B: Convex Sets in Real Infinite-Dimensional Vector Spaces.
520 _aThis textbook is designed for a one-year graduate course in real algebraic geometry, with a particular focus on positivity and sums of squares of polynomials. The first half of the book features a thorough introduction to ordered fields and real closed fields, including the Tarski–Seidenberg projection theorem and transfer principle. Classical results such as Artin's solution to Hilbert's 17th problem and Hilbert's theorems on sums of squares of polynomials are presented in detail. Other features include careful introductions to the real spectrum and to the geometry of semialgebraic sets. The second part studies Archimedean positivstellensätze in great detail and in various settings, together with important applications. The techniques and results presented here are fundamental to contemporary approaches to polynomial optimization. Important results on sums of squares on projective varieties are covered as well. The last part highlights applications to semidefinite programming and polynomial optimization, including recent research on semidefinite representation of convex sets. Written by a leading expert and based on courses taught for several years, the book assumes familiarity with the basics of commutative algebra and algebraic varieties, as can be covered in a one-semester first course. Over 350 exercises, of all levels of difficulty, are included in the book.
650 0 _aAlgebraic geometry.
650 0 _aAlgebraic fields.
_970148
650 0 _aPolynomials.
_964269
650 0 _aAlgebra.
_91501
650 0 _aConvex geometry .
_971929
650 0 _aDiscrete geometry.
_971930
650 0 _aMathematical optimization.
_965522
650 1 4 _aAlgebraic Geometry.
650 2 4 _aField Theory and Polynomials.
_970153
650 2 4 _aOrder, Lattices, Ordered Algebraic Structures.
_969500
650 2 4 _aConvex and Discrete Geometry.
_971931
650 2 4 _aContinuous Optimization.
_977198
710 2 _aSpringerLink (Online service)
773 0 _tSpringer Nature eBook
776 0 8 _iPrinted edition:
_z9783031692123
776 0 8 _iPrinted edition:
_z9783031692147
830 0 _aGraduate Texts in Mathematics,
_x2197-5612 ;
_v303
_972308
856 4 0 _uhttps://doi.org/10.1007/978-3-031-69213-0
912 _aZDB-2-SMA
912 _aZDB-2-SXMS
999 _c63307
_d63307