Real Algebraic Geometry

Here is some basic information about this web page

This web page is under construction.

I created this web page in the fall of 2006. It has pdf files that I wrote for my seminar "Real Algebraic Geometry" at the Institute for Mathematics and its Applications (IMA) at the the University of Minnesota. This seminar began in September, 2006.

You can look at the seminar announcement.

Here is the list of files which are currently available.

• History

  This file mostly contains a time-line of significant events in the 20th century related to Tarksi's work on the decidability of the theory of real closed fields. We discussed this time-line in the seminar on September 14.

• Ordered Rings

  This file contains background information on ordered rings. We discussed this information in the seminar on September 14 and September 28 .

• A First Order Language

  This file contains background information on logic. In particular, it describes the syntax and semantics of a first order language for ordered rings. We discussed this information in the seminar on October 5.

• Closure Operators and Galois Connections

  This file contains some background information on general algebra. In particular, it defines closure operators and Galois connections. It also describes a number of examples of these structures including some which arise in algebraic geometry. We discussed this information in the seminar on October 5.

• Real closed Fields -- Applications of the Intermediate Value Property

  Every real closed field satisfies the intermediate value property. This file contains several applications of this property. These results are used in the proof of Sturm's root counting theorem. We discussed this information in the seminar on October 19.

• Sturm's Root Counting Theorem

  This file contains a proof of Sturm's root counting theorem in the setting of real closed fields. We discussed this information in the seminar on November 2.

• Elimination of Quantifiers

  This file contains the first part of the proof of Tarski's theorem concerning elimnation of quantifiers for real closed fields. We discussed this information in the seminar on November 9 and 16.

• Triangular Toeplitz Matrices

  This file contains a short discussion of triangular Toeplitz matrices. These matrices arise in connection with the division algorithm for polynomials. We discussed part of this information in the seminar on November 23.

• Simply Existential Formulas

  This file contains part of the proof of Tarski's theorem concerning elimnation of quantifiers for real closed fields. We discussed part of this information in the seminar on November 23.

• Definable Sets

  With first order formulas we can associate certain geometric objects called definable sets. This file contains a brief discussion of this association. We discussed part of this information in the seminar on November 30.

• Quadratic Forms

  This file contains a brief discussion of quadratic forms. In particular, we review some of the elementary properties of such forms. We discussed this information in the seminar on December 7 and February 15.

• Hermite's Quadratic Form

  In this section we define Hermite's quadratic form. We also dicuss Hermite's root counting theorem. We discussed this information in the seminar on December 7.

• Decomposition of Algebras Using Idempotents

  This file contains a discussion of the decompostion of finite dimensional algebras using idempotents. It also contains a discussion of several applications of this sort of decomposition. We discussed most of this information in the seminar on February 1.

• Hermite's Quadratic Form Revisited

  In this section we define Hermite's quadratic form and we dicuss Hermite's root counting theorem. We considered this topic in the Fall. We considered it again (in more depth) in the Spring. In particular, we discussed the information in this set of notes in the seminar on February 8 and February 15 .

• Real Rings

  This file contains a discussion of real rings. A ring is "real" if the only sum of squares equal to zero is the trivial one. We discussed this topic in the seminar on March 1 and March 22.

• Cones and Homomorphisms

  This file contains a discussion of the images and inverse images of positive and pre-positive cones under homomorphisms . We discussed this topic in the seminar on March 29.

• Sums of Squares of Real Polynomials

  This file contains a discussion of the connection between sums of squares and positive semi-definite matrices and applications of this connection. We discussed this topic in the seminar on May 10. .

• Sums of Squares of Real Polynomials (continued)

  This file contains further discussion of sums of squares. We discussed this material on May 17.

• Border Basis

  This file contains a discussion of border basis. We discussed this material on May 24 and June 7.

• References

  This file contains a list of refenences.

Last update: June 4, 2007