An introduction to real algebra. (English) Zbl 0577.14016

Lam, T. Y.

Rocky Mt. J. Math. 14, 767-814 (1984).

Very clearly written, with a large amount of comments and motivations, this introduction to real algebra is a fundamental survey on the subject.
The first paragraph is devoted to the level s(A) of a ring A, that is the smallest number n such that -1 can be expressed as a sum of n squares. The second paragraph discusses two notions of reality: semi-reality when \(s(A)=\infty\), and reality when \(\sum a^ 2_ i=0\Rightarrow a_ i=0.\) Paragraph 3 is Artin-Schreier theory for an arbitrary ring: with a convenient definition of ordering (also known as prime cone) of a ring, the result is that a ring is semi-real iff it admits an ordering. The fourth paragraph is about the real spectrum of a ring, that is the space of orderings (or prime cones) of the ring, equipped with a convenient topology and its basic properties. - Paragraph 5 gives an algebraic proof of the Artin-Lang homomorphism theorem via valuation ring techniques, and paragraph 6 contains proofs of Hilbert’s 17th problem and of real nullstellensatz, using the Artin-Lang homomorphism.
Paragraph 7 gives the formal null and positivstellensatz for the real spectrum of an arbitrary ring. In paragraph 8, the beginning steps of the theory of semi-algebraic sets is given. The bijection between constructible sets in the real spectrum of the coordinate ring of a real algebraic set V and the semi-algebraic sets of V appears as a consequence of the Artin-Lang homomorphism theorem. Stengle’s positivstellensätze are then deduced from this bijection and the formal result in paragraph 7.
The paper ends with an historical discussion and comments on the Tarski- Seidenberg principle and the finiteness theorem in semi-algebraic geometry.

Reviewer: M.F.Roy

MSC:

14Pxx	Real algebraic and real-analytic geometry
12D15	Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)

Keywords:

formal nullstellensatz; Hilbert 17th problem; level; semi-reality; Artin- Schreier theory; ordering; prime cone; real spectrum; Artin-Lang homomorphism theorem; real nullstellensatz; positivstellensatz; semi- algebraic sets; Tarski-Seidenberg principle