A Course in Real Algebraic Geometry
Description
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.
Reviews / Votes
"First half of the book develops real algebra, which plays the role in real algebraic geometry that commutative algebra plays in complex algebraic geometry. . The second half of the book presents more advanced material centered around positivity and applications to optimization. . Two appendices provide background from commutative algebra, algebraic geometry, and convexity in infinite-dimensional vector spaces. Each chapter in the book contains a large number of exercises, which makes the book particularly well suited for a graduate course." (Mario Kummer, Mathematical Reviews, December, 2025)
"The book is very well and clearly written. It will serve as a main reference for real algebraic geometry and its applications in polynomial optimization. It is a very valuable source for researchers, teachers and students." (Tobias Kaiser, zbMATH 1559.14001, 2025)
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Person
Claus Scheiderer is Professor for Geometry at Konstanz University (Germany). Among his main mathematical interests are real algebraic geometry, convex algebraic geometry and linear algebraic groups.
Content
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.
