Titlebook: A Course in Real Algebraic Geometry; Positivity and Sums Claus Scheiderer Textbook 2024 The Editor(s) (if applicable) and The Author(s), u

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0072-5285 ing 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 familiarity978-3-031-69213-0Series ISSN 0072-5285 Series E-ISSN 2197-5612

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Anticlimax

The Growth of Synthetic and Imitation Gems,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.

PHAG

https://doi.org/10.1007/978-3-642-67467-9The 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 stellens?tze (of Krivine–Stengle type) are presented, both in an abstract and in a geometric setting.

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Semialgebraic Geometry,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.