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 TarskiSeidenberg 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.

Provides a thorough introduction to modern techniques of real algebraic geometry Contains a detailed account of recent progress on positive polynomials, leading to current research Includes over 350 exercises

Autorentext

Manfred Knebusch is Professor Emeritus at the University of Regensburg. He has written nine books and more than 80 papers on the algebraic theory of quadratic forms over rings and fields, valuation theory, real algebra and real algebraic geometry. His current research focusses on tropical geometry. Claus Scheiderer is Professor at Konstanz University. His primary research interests are real algebraic geometry and convex algebraic geometry. Thomas Unger is Associate Professor at University College Dublin. His research interests include quadratic and hermitian forms, algebras with involution, and noncommutative real algebra and geometry.

Inhalt

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.