Description
2SWS lecture, Wed 10:15 - 11:45 MA 650
Lecturer: Dr. Rosa Preiß
Recommended prior knowledge: Linear Algebra I+II, Algebra I
Algebra II is not a necessary prerequisite! This course can be nicely heard in parallel to Algebra II !
Topics: Ordered fields, ordered field extensions, real closed fields and real closure of an ordered field, semialgebraic sets, Tarski-Seidenberg theorem (polynomial images of semialgebraic sets are semialgebraic), Hilbert's 17th problem (non-negative polynomials), real spectrum of a ring, Positivstellensatz (which polynomials are positive on a given semialgebraic set), Nichtnegativstellensatz, real Nullenstellensatz (which polynomials are zero on a given real algebraic set), abstract versions of the Stellensätze
Literature:
There will be handwritten lecture notes available.
The course is based on the notes of Rainer Sinn
https://www.math.uni-leipzig.de/~sinn/lehre/aufgaben/rag.pdf
and those of Tim Netzer
https://www.uibk.ac.at/mathematik/algebra/media/teaching/ragen.pdf
and the book "Real algebraic geometry" by Bochnak-Coste-Roy.
- Trainer/in: Carlos Enrique Améndola Cerón
- Trainer/in: Rosa Preiß