Mini-Conference Topics in Real Algebraic Geometry

Speaker: Doris Augustin
Title: The membership problem for preorders - the one-dimensional case
Abstract:
The membership problem for a preorder in the polynomial ring over a real closed field is solvable if the set of coefficients of the polynomials which belong to the preorder is weakly semialgebraic; this means: the intersection of this set of coefficients with every finite dimensional subspace of the polynomial ring is semialgebraic. In my talk I will concentrate on finitely generated preorders in the polynomial ring over the field of real numbers in one variable. By describing the structure of preorders generated by one polynomial in the local power series ring at a point we will derive conditions which are necessary and sufficient for a polynomial f to lie in the preorder generated by another polynomial g if the basic closed semialgebraic set generated by g is compact. These conditions imply that the membership problem is solvable for preorders generated by one single polynomial. For finitely generated preorders we are also able to prove that the membership problem is solvable - again by looking at the structure of these preorders in the local power series ring at a point.

Speaker: Ron Brown
Title: Lifting generalized orders on fields to field extensions
Abstract:
We will argue for the utility of Andre Weil's notion of an absolute value on a field which can take the value infinity (ie., an "extended" absolute value) as a central concept of valuation theory. In particular the properties a field Henselian with respect to an extended absolute value and generalized orders and order closures with respect to an extended absolute value will be discussed. (The Henselizations at real extended absolute values are the HRRC fields of Becker, Berr and Gondard, and the closures with respect to p-adic and real extended absolute values are the p-adically closed and real closed fields, respectively.) These notions allow a unification of parts of the theories of formally p-adic and formally real fields, including, for example, generalizations of place existence theorems of Lang and of Prestel and Roquette. Two more recent results wii be presented: an extension existence theorem for generalized orders and a count of the number of extensions of an order on a formally p-adic field to a formally p-adic algebraic extension of that field.

Speaker: Andreas Fischer
Title: John functions in o-minimal structures
Abstract
A John function is a continuously differentiable function $f:U-->R$ such that the product of the norm $\nabla f(x)$ and the distance of $x$ to the complement of $U$ is bounded. We consider the following problem. Given an open subset $U$ of $R^n$. Are there finitely many John functions $f_1,...,f_r$ such that $dist(x,R^n\setminus U)\sum_{i=1}^r\norm{\nabla f_i(x)}$ is also bounded from below by a positive number? If $U$ is definable in some o-minimal structure we can give an affirmative answer, and in this case the $f_i$ are logarithms of definable functions.

Speaker: Pawel Gladki
Title: The pp conjecture for the space of orderings of the field R(x,y).
Abstract:
In this talk we consider the space of orderings $(X_{\R(x,y)}, G_{\R(x,y)})$ of the field of rational functions over $\R$ in two variables. It is shown that the pp conjecture fails to hold for such a space; an example of a simple positive primitive formula which is not product-free and one-related is investigated and it is proven, that although the formula holds true for every finite subspace of $(X_{\R(x,y)}, G_{\R(x,y)})$, it is false in general. This provides a negative answer to one of the questions raised in: M. Marshall, {\em Open questions in the theory of spaces of orderings}, J. Symbolic Logic 67 (2002), 341-352. In a certain sense this work is a continuation of previous results presented in: P. Gladki, M. Marshall, {\em The pp conjecture for spaces of orderings of rational conics}, to appear in J. Algebra Appl., however here new, 'valuation theory free' methods are developed and used.

Speaker: Murray Marshall
Title: New proofs of old results
Abstract:
A new short proof is given for Jacobi's extension of the Kadison-Dubois Theorem. This yields a short uniform proof of bothresults. The Kadison-Dubois Theorem and Jacobi's extension of it are then applied to give short proofs of results of Reznick and of Prestel and Jacobi.

Speaker: Tim Netzer
Title: The Moment Property and Representations of Polynomials.
Abstract:
A finitely generated preordering in the real polynomial ring is said to have the Strong Moment Property, if its closure with respect to the finest locally convex topology equals its saturation. In the literature, there has been introduced a property which implies the Strong Moment Property, known as the "Double-Dagger Property". We discuss an example which shows that this property is strictly stronger than the Strong Moment Property. And unlike the Strong Moment Property, a preordering does not necessarily have it if all preorderings corresponding to the fibers of some bounded polynomials have it.

Speaker: Rajesh Pereira
Title: Matrices over semirings and their applications.
Abstract:
A semiring is a set (R,+,x) with a multiplication and addition which satisfies all of the axioms of a ring with identity except the existence of additive inverses. We give many examples of semirings and then discuss the properties of matrices over semirings including some results and conjectures on diagonally dominant matrices over semirings. (The original portion of this material in this talk is joint work with M. A. Vali).

Speaker: Daniel Plaumann
Title: Sums of squares on reducible real curves
Abstract:
Scheiderer has classified all irreducible real affine curves for which every non-negative regular function is a sum of squares in the coordinate ring. We show how to extend some of these results to reducible curves. We also discuss the moment problem for reducible curves and applications to the moment problem in dimension 2.

Speaker: Marcus Tressl
Title: Weakly semi algebraic orderings and generic extensions of orderings on polynomial rings".
Abstract:
A weakly semi-algebraic ordering of the polynomial ring over a real closed field in several indeterminates is an ordering for which the membership problem is solvable, i.e. for each given degree, the set of coefficient vectors of polynomials contained in the ordering of that degree, is a semi-algebraic set. The Marker-Steinhorn Theorem implies that in the case where R is the real field, every ordering is weakly semi-algebraic. This can be proved by an induction on the dimension, based on the analysis of generic extensions of orderings (so called heirs and coheirs), where the one-dimensional case is trivial. In the talk I will explain this strategy, characterize generic extensions of certain orderings and discuss possible applications to the mebership problem for orderings in the case where R is an arbitrary real closed field.

updated January 21, 2007