04059nam a22006015i 4500 978-3-031-69213-0 DE-He213 20251030113037.0 cr nn 008mamaa 240912s2024 sz | s |||| 0|eng d 9783031692130 978-3-031-69213-0 10.1007/978-3-031-69213-0 doi QA564-609 PBMW bicssc MAT012010 bisacsh PBMW thema Scheiderer, Claus. author. (orcid)0000-0002-5452-1114 https://orcid.org/0000-0002-5452-1114 aut http://id.loc.gov/vocabulary/relators/aut 84157 A Course in Real Algebraic Geometry [electronic resource] : Positivity and Sums of Squares / by Claus Scheiderer. 1st ed. 2024. Cham : Springer International Publishing : Imprint: Springer, 2024. XVIII, 404 p. 5 illus. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Graduate Texts in Mathematics, 2197-5612 ; 303 1 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. This 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. Algebraic geometry. Algebraic fields. 70148 Polynomials. 64269 Algebra. 1501 Convex geometry . 71929 Discrete geometry. 71930 Mathematical optimization. 65522 Algebraic Geometry. Field Theory and Polynomials. 70153 Order, Lattices, Ordered Algebraic Structures. 69500 Convex and Discrete Geometry. 71931 Continuous Optimization. 77198 SpringerLink (Online service) Springer Nature eBook Printed edition: 9783031692123 Printed edition: 9783031692147 Graduate Texts in Mathematics, 2197-5612 ; 303 72308 https://doi.org/10.1007/978-3-031-69213-0 ZDB-2-SMA ZDB-2-SXMS 63307 63307