On a Class of Preorderrings of Higher Level.
On convergence of integrals in o-minimal structures on archimedean real closed fields
We define a notion of volume for sets definable in an o-minimal structure on an archimedean real closed field. We show that given a parametric family of continuous functions on the positive cone of an archimedean real closed field definable in an o-minimal structure, the set of parameters where the integral of the function converges is definable in the same structure.
On Finite Intersections of "Henselian Valued" Fields.
On full cover property of ordered fields
On preordered fields related to Hilbert's 17th problem.
On Real Local-Global Principles.
On Rings admitting orderings and 2-primary chains of orderings of higher level.
On SAP Fields.
On Sign Determinations in Real Algebraic Number Fields.
On Some Hasse Principles over Formally Real Fields.
On the class group of affinoid spaces
On the dimension of the space of ℝ-places of certain rational function fields
We prove that for every n ∈ ℕ the space M(K(x 1, …, x n) of ℝ-places of the field K(x 1, …, x n) of rational functions of n variables with coefficients in a totally Archimedean field K has the topological covering dimension dimM(K(x 1, …, x n)) ≤ n. For n = 2 the space M(K(x 1, x 2)) has covering and integral dimensions dimM(K(x 1, x 2)) = dimℤ M(K(x 1, x 2)) = 2 and the cohomological dimension dimG M(K(x 1, x 2)) = 1 for any Abelian 2-divisible coefficient group G.
On the Extension of Real Places
On the seperation of basic semialgebraic sets by polynomials.
On the theory of cuts in ordered fields.
Ordered fields.
Ordered fields and sign-changing polynomials.
Ordered fields and the ultrafilter theorem
We prove that on the basis of ZF the ultrafilter theorem and the theorem of Artin-Schreier are equivalent. The latter says that every formally real field admits a total order.
Ordered Rings and Fields
We introduce ordered rings and fields following Artin-Schreier’s approach using positive cones. We show that such orderings coincide with total order relations and give examples of ordered (and non ordered) rings and fields. In particular we show that polynomial rings can be ordered in (at least) two different ways [8, 5, 4, 9]. This is the continuation of the development of algebraic hierarchy in Mizar [2, 3].
Ordering Epic R-Fields