2024 | Book

# A Course in Real Algebraic Geometry

## Positivity and Sums of Squares

Author: Claus Scheiderer

Publisher: Springer International Publishing

Book Series : Graduate Texts in Mathematics

2024 | Book

Author: Claus Scheiderer

Publisher: Springer International Publishing

Book Series : Graduate Texts in Mathematics

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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Abstract

Chapter 1 introduces ordered fields and their real closures. The Tarski–Seidenberg theorem is proved and used to present Artin’s solution to Hilbert’s 17th problem. In addition, a proof of the Artin–Lang theorem is given, which characterizes real varieties whose function field can be ordered.

Abstract

Chapter 2 introduces Newton polytopes and Gram matrices of polynomials. After a proof of the Fejér–Riesz theorem, the main topics are Hilbert’s 1888 theorems on the (non-)existence of sum of squares representations of non-negative polynomials.

Abstract

The central object in Chapter 3 is the real spectrum of a ring. The topology of the real spectrum is discussed in some detail, and various stellensätze (of Krivine–Stengle type) are presented, both in an abstract and in a geometric setting.

Abstract

Chapter 4 contains an introduction to central concepts from semialgebraic geometry, such as connected components, dimension or semialgebraic paths. Among the main results are the finiteness theorem and cylindrical algebraic decomposition of semialgebraic sets. Usefulness of the real spectrum is emphasized throughout.

Abstract

The central notion in Chapter 5 is the Archimedean property, the single most important result is the Archimedean positivstellensatz. Major applications are given to polynomials that are strictly positive on some domain, such as the positivstellensätze of Schmüdgen and Putinar. An optional alternative approach is offered, which uses pure states for convex cones and leads to the Archimedean local-global principle.

Abstract

Chapter 6 studies extensions of the strict positivstellensätze from Chapter 5 to non-negative polynomials. The most important tool is the Archimedean local-global principle, which is given an independent second proof. Combining this result with an analysis of (weighted) sums of squares in local rings, a series of existence and non-existence results is obtained for sums of squares representations of non-negative polynomials.

Abstract

Chapter 7 starts with a discussion of varieties of minimal degree. The main result proved in this chapter states that an irreducible projective real variety X has minimal degree if, and only if, every non-negative quadratic form on X is a sum of squares of linear forms. Quantitative refinements are given as well. These results encompass several major classical theorems, among them the Hilbert 1888 theorems that were discussed in Chapter 2.

Abstract

Sums of squares techniques have become an indispensable tool in polynomial and semidefinite optimization. Chapter 8 offers an introduction to some of the most important concepts and results. After an overview of important general notions from convexity, spectrahedra and their shadows are introduced. The moment relaxation approach to polynomial optimization is discussed in detail, it rests on various positivstellensätze from previous chapters. The last part addresses the characterization of spectrahedral shadows, which are the feasible sets of semidefinite programming. Results by Helton–Nie guarantee the existence of semidefinite representations under fairly general conditions. Constructions due to the author show, on the other hand, that prominent convex sets do not allow such a representation.