The contents of the course and other basic information is available in the Student Information System.
The schedule: Monday 10.40am-12.10pm in the seminar room of the Mathematical Institute (to be found also in the Student Information System).
The couse will be about properties of solutions of polynomial equations and inequalities over the real numbers. This is motivated among others by Artin's solution to Hilbert's 17th Problem about real polynomials in multiple indeterminates that attain positive values at all real points.
To explain the solution to Hilbert's problem, one in fact needs more general ordered fields (rational functions over the reals or the Puiseux series for example) as well as the so-called real closed fields and Tarski's quantifier elimintaion, and also the Positivstellensatz (a theorem about the positive locus of a polynomial).
What has been lectured
A brief overview of what has been taught in individual lectures will be updated below.
- October 12, 2015
- Ordered fields, cones, real and real closed fields ([BCR], sec. 1.1-1.2).
- October 26, 2015
- A characterization of real closed fields, analytic properties of polynomial functions ([BCR], sec. 1.2).
- November 2, 2015
- Counting roots of polynomials (Sylvester's and Sturm's theorems), the real closure of an ordered field and its existence ([BCR], sec. 1.2-1.3).
- November 16, 2015
- The uniqueness of the real closure and examples (closures of the fields of rational numbers and of rational functions over the reals), quantifier elimination and the Tarski-Seidenberg principle ([BCR], sec. 1.3-1.4).
- November 23, 2015
- Proof of the Tarski-Seidenberg principle ([BCR], sec. 1.4).
- November 30, 2015
- Algebraic and semialgebraic sets over a real closed field, semialgebraic sets as definable sets, semialgebraic mappings, closed vs. locally closed semialgebraic sets ([BCR], sec. 2.1-2.2).
- December 7, 2015
- Decomposition of semialgebraic sets to open hypercubes, the Artin-Lang homomorphism theorem ([BCR], sec. 2.3 and 4.1).
- December 14, 2015
- Real Nullstellensatz, real ideals of commutative rings, the real radical of an ideal ([BCR], sec. 4.1).
- January 4, 2016
- Cones of commutative rings, P-convex and P-radical ideals, prime cones and their properties ([BCR], sec. 4.2-4.3).
- January 11, 2016
- Positivstellensatz and a solution to Hilbert's 17th Problem ([BCR], sec. 4.4 and 6.1).
The main source for the course is [BCR] below. A more effective algorithm for quantifier elimination from [Col] will be discussed if time permits.
|[BCR]||J. Bochnak, M. Coste, M.-F. Roy, Real algebraic geometry, Springer-Verlag, Berlin, 1998.|
|[Col]||G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Automata theory and formal languages, 134-183, Springer, Berlin, 1975.|