Title of the course: Real algebraic geometry
Instructor: Prof. Max Dickmann
Institution: Paris Diderot U.
Dates: 18-24 September 2017
1. Basic knowledge of algebra and analysis on the real line and Euclidean spaces.
2. Some elementary knowledge of first-order logic will facilitate the ow (and the understanding) of the exposition.
Level: Graduate
LECTURE 1. What is real algebraic geometry? (The distribution of subjects into lectures may be subject to changes.)
  • How it came about (a glimpse into the historical background).
  • How it looks like (with illustrations).
  • Real versus complex algebraic geometry.

LECTURE 2. Ordered fields and real closed fields.

  • Definitions, examples and basic properties.
  • Quantifier elimination for real closed fields, and transfer principles.

LECTURE 3. Applications of quantifier elimination (I).

  • Structure of semi-algebraic sets and functions.
  • The real nullstellensatz.

LECTURE 4. Applications of quantifier elimination (II).

  • Solution of Hilbert's 17th problem.
  • The Artin-Lang homomorphism theorem.
  • Non-singular points of real varieties.

LECTURE 5. Topological structure of semi-algebraic sets. (Subject to time limitation.)

  • Cell decomposition.
  • Open quantifier elimination.
  • Connected components of semi-algebraic sets.


  • J. Bochnak, M. Coste, M.-F. Roy, Real Algebraic Geometry, A Series of Modern Surveys in Mathematics, vol. 36, ix + 430 pp., Springer-Verlag, 1998. [French version; same series and editor, vol. 12, 1987.]
  • M. Dickmann, Applications of Model Theory to Real Algebraic Geometry. A Survey, in Methods in Mathematical Logic. Proceedings, Caracas 1983 (C. Di Prisco, ed.), Lecture Notes Math. 1130, 76 { 150, Springer-Verlag, 1985.

(1) The Bochnak-Coste-Roy book is the most complete (and rather massive) treatise on the subject. In this course it will only be used as a referenceand for some specific items.
(2) My survey is perhaps better adapted to the level and the viewpoint of the course. It also covers subjects that, for lack of time, I will shamelessly omit (e.g., the real spectra of rings).
(3) To be sure, there are other survey articles, written from difierent perspectives, that may be useful references. I cite the following:
* E. Becker, On the real spectrum of a ring and its applications to semi-algebraic geometry, Bulletin Amer. Math. Soc. 15, 19 { 60, 1986.
* M. Knebusch, An invitation to real spectra, in Quadratic and Her-mitian Forms. Conference Hamilton, Ontario, 1983, CMS Conf. Proceedings 4, 5-105, 1984.
* M. Coste An Introduction to Semialgebraic Geometry, Dipartimento di Matematica, Universit_a di Pisa, Italy, 77 pp., 2000.