Overview
- 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
Part of the book series: Graduate Texts in Mathematics (GTM, volume 303)
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About this book
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.
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Table of contents (8 chapters)
Reviews
“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)
Authors and Affiliations
About the author
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.
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Bibliographic Information
Book Title: A Course in Real Algebraic Geometry
Book Subtitle: Positivity and Sums of Squares
Authors: Claus Scheiderer
Series Title: Graduate Texts in Mathematics
DOI: https://doi.org/10.1007/978-3-031-69213-0
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024
Hardcover ISBN: 978-3-031-69212-3Published: 13 September 2024
eBook ISBN: 978-3-031-69213-0Published: 12 September 2024
Series ISSN: 0072-5285
Series E-ISSN: 2197-5612
Edition Number: 1
Number of Pages: XVIII, 404
Number of Illustrations: 5 b/w illustrations
Topics: Algebraic Geometry, Field Theory and Polynomials, Order, Lattices, Ordered Algebraic Structures, Convex and Discrete Geometry, Continuous Optimization